Euler – Coder.si

1 - Multiples of 3 and 5 2 - Even Fibonacci numbers 3 - Largest prime factor 4 - Largest palindrome product 5 - Smallest multiple 6 - Sum square difference 7 - 10001st prime 8 - Largest product in a series 9 - Special Pythagorean triplet 10 - Summation of primes 11 - Largest product in a grid 12 - Highly divisible triangular number 13 - Large sum 14 - Longest Collatz sequence 15 - Lattice paths 16 - Power digit sum 17 - Number letter counts 18 - Maximum path sum I 19 - Counting Sundays 20 - Factorial digit sum 21 - Amicable numbers 22 - Names scores 23 - Non-abundant sums 24 - Lexicographic permutations 25 - 1000-digit Fibonacci number 26 - Reciprocal cycles 27 - Quadratic primes 28 - Number spiral diagonals 29 - Distinct powers 30 - Digit fifth powers 31 - Coin sums 32 - Pandigital products 33 - Digit cancelling fractions 34 - Digit factorials 35 - Circular primes 36 - Double-base palindromes 37 - Truncatable primes 38 - Pandigital multiples 39 - Integer right triangles 40 - Champernowne's constant 41 - Pandigital prime 42 - Coded triangle numbers 43 - Sub-string divisibility 44 - Pentagon numbers 45 - Triangular, pentagonal, and hexagonal 46 - Goldbach's other conjecture 47 - Distinct primes factors 48 - Self powers 49 - Prime permutations 50 - Consecutive prime sum 51 - Prime digit replacements 52 - Permuted multiples 53 - Combinatoric selections 54 - Poker hands 55 - Lychrel numbers 56 - Powerful digit sum 57 - Square root convergents 58 - Spiral primes 59 - XOR decryption 60 - Prime pair sets 61 - Cyclical figurate numbers 62 - Cubic permutations 63 - Powerful digit counts 64 - Odd period square roots 65 - Convergents of e 66 - Diophantine equation 67 - Maximum path sum II 68 - Magic 5-gon ring 69 - Totient maximum 70 - Totient permutation 71 - Ordered fractions 72 - Counting fractions 73 - Counting fractions in a range 74 - Digit factorial chains 75 - Singular integer right triangles 76 - Counting summations 77 - Prime summations 78 - Coin partitions 79 - Passcode derivation 80 - Square root digital expansion 81 - Path sum: two ways 82 - Path sum: three ways 83 - Path sum: four ways 84 - Monopoly odds 85 - Counting rectangles 86 - Cuboid route 87 - Prime power triples 88 - Product-sum numbers 89 - Roman numerals 90 - Cube digit pairs 91 - Right triangles with integer coordinates 92 - Square digit chains 93 - Arithmetic expressions 94 - Almost equilateral triangles 95 - Amicable chains 96 - Su Doku 97 - Large non-Mersenne prime 98 - Anagramic squares 99 - Largest exponential 100 - Arranged probability 101 - Optimum polynomial 102 - Triangle containment 103 - Special subset sums: optimum 104 - Pandigital Fibonacci ends 105 - Special subset sums: testing 106 - Special subset sums: meta-testing 107 - Minimal network 108 - Diophantine reciprocals I 109 - Darts 110 - Diophantine reciprocals II 111 - Primes with runs 112 - Bouncy numbers 113 - Non-bouncy numbers 114 - Counting block combinations I 115 - Counting block combinations II 116 - Red, green or blue tiles 117 - Red, green, and blue tiles 118 - Pandigital prime sets 119 - Digit power sum 120 - Square remainders 121 - Disc game prize fund 122 - Efficient exponentiation 123 - Prime square remainders 124 - Ordered radicals 125 - Palindromic sums 126 - Cuboid layers 127 - abc-hits 128 - Hexagonal tile differences 129 - Repunit divisibility 130 - Composites with prime repunit property 131 - Prime cube partnership 132 - Large repunit factors 133 - Repunit nonfactors 134 - Prime pair connection 135 - Same differences 136 - Singleton difference 137 - Fibonacci golden nuggets 138 - Special isosceles triangles 139 - Pythagorean tiles 140 - Modified Fibonacci golden nuggets 141 - Investigating progressive numbers, n, which are also square 142 - Perfect Square Collection 143 - Investigating the Torricelli point of a triangle 144 - Investigating multiple reflections of a laser beam 145 - How many reversible numbers are there below one-billion?146 - Investigating a Prime Pattern 147 - Rectangles in cross-hatched grids 148 - Exploring Pascal's triangle 149 - Searching for a maximum-sum subsequence 150 - Searching a triangular array for a sub-triangle having minimum-sum 151 - Paper sheets of standard sizes: an expected-value problem 152 - Writing 1/2 as a sum of inverse squares 153 - Investigating Gaussian Integers 154 - Exploring Pascal's pyramid 155 - Counting Capacitor Circuits 156 - Counting Digits 157 - Solving the diophantine equation 1/a+1/b= p/10n 158 - Exploring strings for which only one character comes lexicographically after its neighbour to the left 159 - Digital root sums of factorisations 160 - Factorial trailing digits 161 - Triominoes 162 - Hexadecimal numbers 163 - Cross-hatched triangles 164 - Numbers for which no three consecutive digits have a sum greater than a given value 165 - Intersections 166 - Criss Cross 167 - Investigating Ulam sequences 168 - Number Rotations 169 - Exploring the number of different ways a number can be expressed as a sum of powers of 2 170 - Find the largest 0 to 9 pandigital that can be formed by concatenating products 171 - Finding numbers for which the sum of the squares of the digits is a square 172 - Investigating numbers with few repeated digits 173 - Using up to one million tiles how many different "hollow" square laminae can be formed?174 - Counting the number of "hollow" square laminae that can form one, two, three, ... distinct arrangements 175 - Fractions involving the number of different ways a number can be expressed as a sum of powers of 2 176 - Right-angled triangles that share a cathetus 177 - Integer angled Quadrilaterals 178 - Step Numbers 179 - Consecutive positive divisors 180 - Rational zeros of a function of three variables 181 - Investigating in how many ways objects of two different colours can be grouped 182 - RSA encryption 183 - Maximum product of parts 184 - Triangles containing the origin 185 - Number Mind 186 - Connectedness of a network 187 - Semiprimes 188 - The hyperexponentiation of a number 189 - Tri-colouring a triangular grid 190 - Maximising a weighted product 191 - Prize Strings 192 - Best Approximations 193 - Squarefree Numbers 194 - Coloured Configurations 195 - Inscribed circles of triangles with one angle of 60 degrees 196 - Prime triplets 197 - Investigating the behaviour of a recursively defined sequence 198 - Ambiguous Numbers 199 - Iterative Circle Packing 200 - Find the 200th prime-proof sqube containing the contiguous sub-string "200"201 - Subsets with a unique sum 202 - Laserbeam 203 - Squarefree Binomial Coefficients 204 - Generalised Hamming Numbers 205 - Dice Game 206 - Concealed Square 207 - Integer partition equations 208 - Robot Walks 209 - Circular Logic 210 - Obtuse Angled Triangles 211 - Divisor Square Sum 212 - Combined Volume of Cuboids 213 - Flea Circus 214 - Totient Chains 215 - Crack-free Walls 216 - Investigating the primality of numbers of the form 2n2-1 217 - Balanced Numbers 218 - Perfect right-angled triangles 219 - Skew-cost coding 220 - Heighway Dragon 221 - Alexandrian Integers 222 - Sphere Packing 223 - Almost right-angled triangles I 224 - Almost right-angled triangles II 225 - Tribonacci non-divisors 226 - A Scoop of Blancmange 227 - The Chase 228 - Minkowski Sums 229 - Four Representations using Squares 230 - Fibonacci Words 231 - The prime factorisation of binomial coefficients 232 - The Race 233 - Lattice points on a circle 234 - Semidivisible numbers 235 - An Arithmetic Geometric sequence 236 - Luxury Hampers 237 - Tours on a 4 x n playing board 238 - Infinite string tour 239 - Twenty-two Foolish Primes 240 - Top Dice 241 - Perfection Quotients 242 - Odd Triplets 243 - Resilience 244 - Sliders 245 - Coresilience 246 - Tangents to an ellipse 247 - Squares under a hyperbola 248 - Numbers for which Euler’s totient function equals 13!249 - Prime Subset Sums 250 - 250250 251 - Cardano Triplets 252 - Convex Holes 253 - Tidying up 254 - Sums of Digit Factorials 255 - Rounded Square Roots 256 - Tatami-Free Rooms 257 - Angular Bisectors 258 - A lagged Fibonacci sequence 259 - Reachable Numbers 260 - Stone Game 261 - Pivotal Square Sums 262 - Mountain Range 263 - An engineers' dream come true 264 - Triangle Centres 265 - Binary Circles 266 - Pseudo Square Root 267 - Billionaire 268 - Counting numbers with at least four distinct prime factors less than 100 269 - Polynomials with at least one integer root 270 - Cutting Squares 271 - Modular Cubes, part 1 272 - Modular Cubes, part 2 273 - Sum of Squares 274 - Divisibility Multipliers 275 - Balanced Sculptures 276 - Primitive Triangles 277 - A Modified Collatz sequence 278 - Linear Combinations of Semiprimes 279 - Triangles with integral sides and an integral angle 280 - Ant and seeds 281 - Pizza Toppings 282 - The Ackermann function 283 - Integer sided triangles for which the area/perimeter ratio is integral 284 - Steady Squares 285 - Pythagorean odds 286 - Scoring probabilities 287 - Quadtree encoding (a simple compression algorithm)288 - An enormous factorial 289 - Eulerian Cycles 290 - Digital Signature 291 - Panaitopol Primes 292 - Pythagorean Polygons 293 - Pseudo-Fortunate Numbers 294 - Sum of digits - experience #23 295 - Lenticular holes 296 - Angular Bisector and Tangent 297 - Zeckendorf Representation 298 - Selective Amnesia 299 - Three similar triangles 300 - Protein folding 301 - Nim 302 - Strong Achilles Numbers 303 - Multiples with small digits 304 - Primonacci 305 - Reflexive Position 306 - Paper-strip Game 307 - Chip Defects 308 - An amazing Prime-generating Automaton 309 - Integer Ladders 310 - Nim Square 311 - Biclinic Integral Quadrilaterals 312 - Cyclic paths on Sierpiński graphs 313 - Sliding game 314 - The Mouse on the Moon 315 - Digital root clocks 316 - Numbers in decimal expansions 317 - Firecracker 318 - 2011 nines 319 - Bounded Sequences 320 - Factorials divisible by a huge integer 321 - Swapping Counters 322 - Binomial coefficients divisible by 10 323 - Bitwise-OR operations on random integers 324 - Building a tower 325 - Stone Game II 326 - Modulo Summations 327 - Rooms of Doom 328 - Lowest-cost Search 329 - Prime Frog 330 - Euler's Number 331 - Cross flips 332 - Spherical triangles 333 - Special partitions 334 - Spilling the beans 335 - Gathering the beans 336 - Maximix Arrangements 337 - Totient Stairstep Sequences 338 - Cutting Rectangular Grid Paper 339 - Peredur fab Efrawg 340 - Crazy Function 341 - Golomb's self-describing sequence 342 - The totient of a square is a cube 343 - Fractional Sequences 344 - Silver dollar game 345 - Matrix Sum 346 - Strong Repunits 347 - Largest integer divisible by two primes 348 - Sum of a square and a cube 349 - Langton's ant 350 - Constraining the least greatest and the greatest least 351 - Hexagonal orchards 352 - Blood tests 353 - Risky moon 354 - Distances in a bee's honeycomb 355 - Maximal coprime subset 356 - Largest roots of cubic polynomials 357 - Prime generating integers 358 - Cyclic numbers 359 - Hilbert's New Hotel 360 - Scary Sphere 361 - Subsequence of Thue-Morse sequence 362 - Squarefree factors 363 - Bézier Curves 364 - Comfortable distance 365 - A huge binomial coefficient 366 - Stone Game III 367 - Bozo sort 368 - A Kempner-like series 369 - Badugi 370 - Geometric triangles 371 - Licence plates 372 - Pencils of rays 373 - Circumscribed Circles 374 - Maximum Integer Partition Product 375 - Minimum of subsequences 376 - Nontransitive sets of dice 377 - Sum of digits, experience 13 378 - Triangle Triples 379 - Least common multiple count 380 - Amazing Mazes!381 - (prime-k) factorial 382 - Generating polygons 383 - Divisibility comparison between factorials 384 - Rudin-Shapiro sequence 385 - Ellipses inside triangles 386 - Maximum length of an antichain 387 - Harshad Numbers 388 - Distinct Lines 389 - Platonic Dice 390 - Triangles with non rational sides and integral area 391 - Hopping Game 392 - Enmeshed unit circle 393 - Migrating ants 394 - Eating pie 395 - Pythagorean tree 396 - Weak Goodstein sequence 397 - Triangle on parabola 398 - Cutting rope 399 - Squarefree Fibonacci Numbers 400 - Fibonacci tree game 401 - Sum of squares of divisors 402 - Integer-valued polynomials 403 - Lattice points enclosed by parabola and line 404 - Crisscross Ellipses 405 - A rectangular tiling 406 - Guessing Game 407 - Idempotents 408 - Admissible paths through a grid 409 - Nim Extreme 410 - Circle and tangent line 411 - Uphill paths 412 - Gnomon numbering 413 - One-child Numbers 414 - Kaprekar constant 415 - Titanic sets 416 - A frog's trip 417 - Reciprocal cycles II 418 - Factorisation triples 419 - Look and say sequence 420 - 2x2 positive integer matrix 421 - Prime factors of n15+1 422 - Sequence of points on a hyperbola 423 - Consecutive die throws 424 - Kakuro 425 - Prime connection 426 - Box-ball system 427 - n-sequences 428 - Necklace of circles 429 - Sum of squares of unitary divisors 430 - Range flips 431 - Square Space Silo 432 - Totient sum 433 - Steps in Euclid's algorithm 434 - Rigid graphs 435 - Polynomials of Fibonacci numbers 436 - Unfair wager 437 - Fibonacci primitive roots 438 - Integer part of polynomial equation's solutions 439 - Sum of sum of divisors 440 - GCD and Tiling 441 - The inverse summation of coprime couples 442 - Eleven-free integers 443 - GCD sequence 444 - The Roundtable Lottery 445 - Retractions A 446 - Retractions B 447 - Retractions C 448 - Average least common multiple 449 - Chocolate covered candy 450 - Hypocycloid and Lattice points 451 - Modular inverses 452 - Long Products 453 - Lattice Quadrilaterals 454 - Diophantine reciprocals III 455 - Powers With Trailing Digits 456 - Triangles containing the origin II 457 - A polynomial modulo the square of a prime 458 - Permutations of Project 459 - Flipping game 460 - An ant on the move 461 - Almost Pi 462 - Permutation of 3-smooth numbers 463 - A weird recurrence relation 464 - Möbius function and intervals 465 - Polar polygons 466 - Distinct terms in a multiplication table 467 - Superinteger 468 - Smooth divisors of binomial coefficients 469 - Empty chairs 470 - Super Ramvok 471 - Triangle inscribed in ellipse 472 - Comfortable Distance II 473 - Phigital number base 474 - Last digits of divisors 475 - Music festival 476 - Circle Packing II 477 - Number Sequence Game 478 - Mixtures 479 - Roots on the Rise 480 - The Last Question 481 - Chef Showdown 482 - The incenter of a triangle 483 - Repeated permutation 484 - Arithmetic Derivative 485 - Maximum number of divisors 486 - Palindrome-containing strings 487 - Sums of power sums 488 - Unbalanced Nim 489 - Common factors between two sequences 490 - Jumping frog 491 - Double pandigital number divisible by 11 492 - Exploding sequence 493 - Under The Rainbow 494 - Collatz prefix families 495 - Writing n as the product of k distinct positive integers 496 - Incenter and circumcenter of triangle 497 - Drunken Tower of Hanoi 498 - Remainder of polynomial division 499 - St. Petersburg Lottery 500 - Problem 500!!!501 - Eight Divisors 502 - Counting Castles 503 - Compromise or persist 504 - Square on the Inside 505 - Bidirectional Recurrence 506 - Clock sequence 507 - Shortest Lattice Vector 508 - Integers in base i-1 509 - Divisor Nim 510 - Tangent Circles 511 - Sequences with nice divisibility properties 512 - Sums of totients of powers 513 - Integral median 514 - Geoboard Shapes 515 - Dissonant Numbers 516 - 5-smooth totients 517 - A real recursion 518 - Prime triples and geometric sequences 519 - Tricoloured Coin Fountains 520 - Simbers 521 - Smallest prime factor 522 - Hilbert's Blackout 523 - First Sort I 524 - First Sort II 525 - Rolling Ellipse 526 - Largest prime factors of consecutive numbers 527 - Randomized Binary Search 528 - Constrained Sums 529 - 10-substrings 530 - GCD of Divisors 531 - Chinese leftovers 532 - Nanobots on Geodesics 533 - Minimum values of the Carmichael function 534 - Weak Queens 535 - Fractal Sequence 536 - Modulo power identity 537 - Counting tuples 538 - Maximum quadrilaterals 539 - Odd elimination 540 - Counting primitive Pythagorean triples 541 - Divisibility of Harmonic Number Denominators 542 - Geometric Progression with Maximum Sum 543 - Prime-Sum Numbers 544 - Chromatic Conundrum 545 - Faulhaber's Formulas 546 - The Floor's Revenge 547 - Distance of random points within hollow square laminae 548 - Gozinta Chains 549 - Divisibility of factorials 550 - Divisor game 551 - Sum of digits sequence 552 - Chinese leftovers II 553 - Power sets of power sets 554 - Centaurs on a chess board 555 - McCarthy 91 function 556 - Squarefree Gaussian Integers 557 - Cutting triangles 558 - Irrational base 559 - Permuted Matrices 560 - Coprime Nim 561 - Divisor Pairs 562 - Maximal perimeter 563 - Robot Welders 564 - Maximal polygons 565 - Divisibility of sum of divisors 566 - Cake Icing Puzzle 567 - Reciprocal games I 568 - Reciprocal games II 569 - Prime Mountain Range 570 - Snowflakes 571 - Super Pandigital Numbers 572 - Idempotent matrices 573 - Unfair race 574 - Verifying Primes 575 - Wandering Robots 576 - Irrational jumps 577 - Counting hexagons 578 - Integers with decreasing prime powers 579 - Lattice points in lattice cubes 580 - Squarefree Hilbert numbers 581 - 47-smooth triangular numbers 582 - Nearly isosceles 120 degree triangles 583 - Heron Envelopes 584 - Birthday Problem Revisited 585 - Nested square roots 586 - Binary Quadratic Form 587 - Concave triangle 588 - Quintinomial coefficients 589 - Poohsticks Marathon 590 - Sets with a given Least Common Multiple 591 - Best Approximations by Quadratic Integers 592 - Factorial trailing digits 2 593 - Fleeting Medians 594 - Rhombus Tilings 595 - Incremental Random Sort 596 - Number of lattice points in a hyperball 597 - Torpids 598 - Split Divisibilities 599 - Distinct Colourings of a Rubik's Cube 600 - Integer sided equiangular hexagons 601 - Divisibility streaks 602 - Product of Head Counts 603 - Substring sums of prime concatenations 604 - Convex path in square 605 - Pairwise Coin-Tossing Game 606 - Gozinta Chains II 607 - Marsh Crossing 608 - Divisor Sums 609 - π sequences 610 - Roman Numerals II 611 - Hallway of square steps 612 - Friend numbers 613 - Pythagorean Ant 614 - Special partitions 2 615 - The millionth number with at least one million prime factors 616 - Creative numbers 617 - Mirror Power Sequence 618 - Numbers with a given prime factor sum 619 - Square subsets 620 - Planetary Gears 621 - Expressing an integer as the sum of triangular numbers 622 - Riffle Shuffles 623 - Lambda Count 624 - Two heads are better than one 625 - Gcd sum 626 - Counting Binary Matrices 627 - Counting products 628 - Open chess positions 629 - Scatterstone Nim 630 - Crossed lines 631 - Constrained Permutations 632 - Square prime factors 633 - Square prime factors II 634 - Numbers of the form $a^2b^3$635 - Subset sums 636 - Restricted Factorisations 637 - Flexible digit sum 638 - Weighted lattice paths 639 - Summing a multiplicative function 640 - Shut the Box 641 - A Long Row of Dice 642 - Sum of largest prime factors 643 - 2-Friendly 644 - Squares on the line 645 - Every Day is a Holiday 646 - Bounded Divisors 647 - Linear Transformations of Polygonal Numbers 648 - Skipping Squares 649 - Low-Prime Chessboard Nim 650 - Divisors of Binomial Product 651 - Patterned Cylinders 652 - Distinct values of a proto-logarithmic function 653 - Frictionless Tube 654 - Neighbourly Constraints 655 - Divisible Palindromes 656 - Palindromic sequences 657 - Incomplete words 658 - Incomplete words II 659 - Largest prime 660 - Pandigital Triangles 661 - A Long Chess Match 662 - Fibonacci paths 663 - Sums of subarrays 664 - An infinite game 665 - Proportionate Nim 666 - Polymorphic Bacteria 667 - Moving Pentagon 668 - Square root smooth Numbers 669 - The King's Banquet 670 - Colouring a Strip 671 - Colouring a Loop 672 - One more one 673 - Beds and Desks 674 - Solving $\mathcal{I}$-equations 675 - $2^{\omega(n)}$676 - Matching Digit Sums 677 - Coloured Graphs 678 - Fermat-like Equations 679 - Freefarea 680 - Yarra Gnisrever 681 - Maximal Area 682 - 5-Smooth Pairs 683 - The Chase II 684 - Inverse Digit Sum 685 - Inverse Digit Sum II 686 - Powers of Two 687 - Shuffling Cards 688 - Piles of Plates 689 - Binary Series 690 - Tom and Jerry 691 - Long substring with many repetitions 692 - Siegbert and Jo 693 - Finite Sequence Generator 694 - Cube-full Divisors

← Problem 504

Bidirectional Recurrence

Problem 506 →

Bidirectional Recurrence

Problem 505

Let:

You are given: