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Message: <blockquote data-quote="MaD-DoC" data-source="post: 2312703" data-attributes="member: 56284">500 is the number of planar partitions of 10.   501 is the number of partitions of 5 items into ordered lists.   502 uses the same digits as φ(502).   503 is the smallest prime which is the sum of the cubes of the first few primes.   504 = 9P3.   505 = 10C5 + 10C0 + 10C5.   506 is the sum of the first 11 squares.   507 is the number of rooted ternary trees with 10 vertices.   508 ???   509 is the index of a prime Fibonacci number.   510 is the number of binary rooted trees with 14 vertices.   511 = 111111111 in base 2.   512 is the cube of the sum of its digits.   513 is the number of conjugacy classes of the alternating group A22.   514 ???   515 is the number of graphs on 6 vertices with no isolated vertices.   516 is the number of partitions of 32 in which no part occurs only once.   517 does not occur in its factorial in base 2.   518 = 51 + 12 + 83.   519 is the number of trees on 15 vertices with diameter 5.   520 is the number of ways to place 2 non-attacking kings on a 6×6 chessboard.   521 is the 13th Lucas number.   522 is the number of ways to place a non-attacking white and black pawn on a 6×6 chessboard.   523 is the smallest prime that is followed by 17 composite numbers.   524 is the number of 6-kings.   525 is a hexagonal pyramidal number. 526 is the number of ways to cut a 8&rimes;8 chessboard into 2 pieces with equal areas with a cut that only travels up and right.   527 is the smallest number n for which there do not exist 4 smaller numbers so that a1! a2! a3! a4! n! is square.   528 concatenated with its successor is square.   529 is the smallest number n so that the continued fraction for n/k contains no 2's for any 1 ≤ k ≤ n.   530 is the sum of the first 3 perfect numbers.   531 is the smallest number with the property that its first 4 multiples contain the digit 1.   532 is a hendecagonal pyramidal number.   533 is the number of degree sequences for graphs with 5 vertices.   534 ???   535 is a palindrome whose φ<img src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" class="smilie smilie--sprite smilie--sprite23" alt="(n)" title="Thumbs down    (n)" loading="lazy" data-shortname="(n)" /> is also palindromic.   536 is the number of solutions of the stomachion puzzle.   537 divides the sum of the cubes of the first 537 primes.   538 is the 10th open meandric number.   539 is the number of multigraphs with 5 vertices and 9 edges.   540 is divisible by its reverse.   541 is the number of orderings of 5 objects with ties allowed.   542 is a member of the Fibonacci-type sequence starting with 3 and 8.   543 is a number whose square and cube use different digits.   544 is the generalized Catalan number C(14,3).   545 has a base 3 representation that begins with its base 4 representation.   546 undulates in bases 3, 4, and 5.   547 is the smallest number that can not be written using 11 copies of 11 and the operations +, –, ×, and ÷.   548 is the maximum number of 9th powers needed to sum to any number.   549 ???   550 is a pentagonal pyramidal number.   551 is the number of trees with 12 vertices.   552 is the number of prime knots with 11 crossings.   553 is a Huay rhombic dodecahedral number.   554 is the number of self-dual planar graphs with 20 edges.   555 is a repdigit.   556 are the first 3 digits of 4556.   557 ???   558 divides the sum of the largest prime factors of the first 558 positive integers.   559 is a centered cube number.   560 = 16C3.   561 is the smallest Carmichael number.   562 is the maximum number of regions a circle can be cut into by joining 11 points on the circumference with straight lines.   563 is the largest known Wilson prime.   564 is the number of 13-ominoes with a horizontal or vertical line of symmetry.   565 is a structured truncated octahedral number.   566 is the number of ways to place 24 points on a 12×12 grid so that no 3 points are on a line.   567 has the property that it and its square together use the digits 1-9 once.   568 is the smallest number whose 7th power can be written as the sum of seven 7th powers.   569 is the smallest number n for which the concatenation of n, (n+1), ... (n+30) is prime.   570 is the product of all the prime palindromic Roman numerals.   571 is the index of a prime Fibonacci number.   572 is the smallest number which has equal numbers of every digit in bases 2 and 3.   573 has the property that its square is the concatenation of two consecutive numbers.   574 is the maximum number of pieces a torus can be cut into with 14 cuts.   575 is a palindrome that is one less than a square.   576 is the number of 4×4 Latin squares.   577 is a Proth prime.   578 is the number of graphs with 7 vertices with clique number 3.   579 is the number of graphs with 7 vertices that have chromatic number 3.   580 is the 6th central quadrinomial coefficient.   581 has a base 3 representation that begins with its base 4 representation.   582 is the number of antisymmetric relations on a 5 element set.   583 is the smallest number whose reciprocal has period 26.   584 is the number of ways to color the vertices of a triangle with 12 colors, up to rotation.   585 is a palindrome in base 2, base 8, and in base 10.   586 is the smallest number that appears in its factorial 6 times.   587 is the smallest number whose digit sum is larger than that of its cube.   588 is the number of possible rook moves on a 7×7 chessboard.   589 is a centered tetrahedral number.   590 is a value of n for which φ<img src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" class="smilie smilie--sprite smilie--sprite23" alt="(n)" title="Thumbs down    (n)" loading="lazy" data-shortname="(n)" /> + φ(n+1) divides σ<img src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" class="smilie smilie--sprite smilie--sprite23" alt="(n)" title="Thumbs down    (n)" loading="lazy" data-shortname="(n)" /> + σ(n+1).   591 is the number of ways to stack 23 boxes in a line so that each box lies on the table or on a box next to 2 boxes.   592 evenly divides the sum of its rotations.   593 is a Leyland number.   594 = 15 + 29 + 34.   595 is the number of ways to tile a 3×18 rectangle with 3×1 rectangles.   596 is the number of Hamiltonian cycles of a 4×9 rectangle graph.   597 is a value of n for which n!!! + 1 is prime.   598 = 51 + 92 + 83.   599 is the smallest number whose digits add to 23.   600 and its reverse are both the averages of twin primes.   601 is the location of the first occurrence of 3 consecutive zeroes in the decimal digits of π.   602 are the first 3 digits of 5602.   603 is the smallest number n so that n, n+1, and n+2 are all the product of a prime and the square of a prime.   604 and the two numbers before it and after it are all products of exactly 3 primes.   605 has a sum of digits equal to its largest prime factor.   606 is the first non-trivial number that is both 11-gonal and centered 11-gonal.   607 is the exponent of a Mersenne prime.   608 is a number that does not have any digits in common with its cube.   609 is a strobogrammatic number.   610 is the smallest Fibonacci number that begins with 6.   611 ???   612 is a number whose square and cube use different digits.   613 is the index of a prime Lucas number.   614 is the smallest number that can be written as the sum of 3 squares in 9 ways.   615 is the trinomial coefficient T(10,6).   616 is a Padovan number.   617 = 1!2 + 2!2 + 3!2 + 4!2.   618 is the number of ternary square-free words of length 15.   619 is a strobogrammatic prime.   620 is the number of sided 7-hexes.   621 is the number of ways to 9-color the faces of a tetrahedron.   622 ???   623 is the number of inequivalent asymmetric Ferrers graphs with 23 points.   624 is the smallest number with the property that its first 5 multiples contain the digit 2.   625 is an automorphic number.   626 is a palindrome in base 5 and in base 10.   627 is the number of partitions of 20.   628 is the sum of the squares of 4 consecutive primes.   629 evenly divides the sum of its rotations.   630 is a triangular number, 3 times a triangular number, and 6 times a triangular number.   631 has a base 2 representation that begins with its base 5 representation.   632 is the number of necklaces (that can't be turned over) possible with 13 beads, each being one of 2 colors.   633 is the smallest number n whose 5th root has a decimal part that begins with the digits of n.   634 is a number n whose 5th root has a decimal part that begins with the digits of n.   635 is a number n whose 5th root has a decimal part that begins with the digits of n.   636 is a number n whose 5th root has a decimal part that begins with the digits of n.   637 = 777 in base 9.   638 is the number of fixed 5-kings.   639 is a number n whose 5th root has a decimal part that begins with the digits of n.   640 = 16!!!!!!.   641 is the smallest prime factor of 225+1.   642 is the smallest number with the property that its first 6 multiples contain the digit 2.   643 is the largest prime factor of 123456.   644 is a Perrin number.   645 is the largest n for which 1+2+3+ ... +n = 12+22+32+ ... +k2 for some k.   646 is the number of connected planar graphs with 7 vertices.   647 ???   648 is the smallest number whose decimal part of its 6th root begins with the digits 1-9 in some order.   650 is the sum of the first 12 squares.   651 has a 4th power that is the sum of four 4th powers.   652 is the only known non-perfect number whose number of divisors and sum of smaller divisors are perfect.   653 is the only known prime for which 5 is neither a primitive root or a quadratic residue of 4n2+1.   654 has a square that is the sum of a cube and 5th power.   655 ???   656 is a palindrome in base 3 and in base 10.   657 is the number of ways to tile a 4×22 rectangle with 4×1 rectangles.   658 is the number of triangles of any size contained in the triangle of side 13 on a triangular grid.   659 is an Eisenstein-Mersenne prime.   660 is the order of a non-cyclic simple group.   661 is the largest prime factor of 8! + 1.   662 is the index of the smallest triangular number that contains the digits 1, 2, 3, 4, and 5.   663 is the generalized Catalan number C(15,3).   664 is a value of n so that n(n+7) is a palindrome.   665 is a member of the Fibonacci-type sequence starting with 1 and 4.   666 is the largest rep-digit triangular number.   667 is the number of asymmetric trees with 16 vertices.   668 is the number of legal pawn moves in chess.   669 is the number of unsymmetrical ways to dissect a regular 12-gon into 10 triangles.   670 is an octahedral number.   671 is a rhombic dodecahedral number.   672 is a multi-perfect number.   673 is a tetranacci number.   674 ???   675 is the smallest order for which there are 17 groups.   676 is the smallest palindromic square number whose square root is not palindromic.   677 is the closest integer to 11e.   678 is a member of the Fibonacci-type sequence starting with 1 and 7.   679 is the smallest number with multiplicative persistence 5.   680 is the smallest tetrahedral number that is also the sum of 2 tetrahedral numbers.   681 divides the sum of the first 681 composite numbers.   682 = 11C6 + 11C8 + 11C2.   683 is a Wagstaff prime.   684 is the sum of 3 consecutive cubes.   685 ???   686 is the number of partitions of 35 in which no part occurs only once.   687 is the closest integer to 8π.   688 is a Friedman number.   689 is the smallest number that can be written as the sum of 3 distinct squares in 9 ways.   690 is the smallest number that can not be written as the sum of a triangular number, a cube, and a Fibonacci number.   691 is the smallest prime p for which x5 = x4 + x3 + x2 + x + 1 (mod p) has 5 solutions.   692 is a number that does not have any digits in common with its cube.   693 are the first 3 decimal digits of ln(2).   694 is the number of different arrangements (up to rotation and reflection) of 7 non-attacking rooks on a 7×7 chessboard.   695 is the maximum number of pieces a torus can be cut into with 15 cuts.   696 is a palindrome n so that n(n+is also palindromic.   697 is a 12-hyperperfect number.   698 ???   699 is a value of n for which |cos<img src="data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7" class="smilie smilie--sprite smilie--sprite23" alt="(n)" title="Thumbs down    (n)" loading="lazy" data-shortname="(n)" />| is smaller than any previous integer.</blockquote>

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