This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A340603 #12 Jan 22 2021 20:28:51 %S A340603 3,4,7,10,12,13,15,16,18,19,22,25,27,28,29,33,34,37,40,42,43,46,48,51, %T A340603 52,53,55,60,61,62,63,64,69,70,71,72,76,77,78,79,82,85,88,89,90,93,94, %U A340603 98,100,101,105,107,108,112,113,114,115,116,117,118,119,121 %N A340603 Heinz numbers of integer partitions of odd rank. %C A340603 The Dyson rank of a nonempty partition is its maximum part minus its number of parts. The rank of an empty partition is 0. %C A340603 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions. %H A340603 FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000145">St000145: The Dyson rank of a partition</a> %F A340603 A061395(a(n)) - A001222(a(n)) is odd. %e A340603 The sequence of partitions with their Heinz numbers begins: %e A340603 3: (2) 33: (5,2) 63: (4,2,2) %e A340603 4: (1,1) 34: (7,1) 64: (1,1,1,1,1,1) %e A340603 7: (4) 37: (12) 69: (9,2) %e A340603 10: (3,1) 40: (3,1,1,1) 70: (4,3,1) %e A340603 12: (2,1,1) 42: (4,2,1) 71: (20) %e A340603 13: (6) 43: (14) 72: (2,2,1,1,1) %e A340603 15: (3,2) 46: (9,1) 76: (8,1,1) %e A340603 16: (1,1,1,1) 48: (2,1,1,1,1) 77: (5,4) %e A340603 18: (2,2,1) 51: (7,2) 78: (6,2,1) %e A340603 19: (8) 52: (6,1,1) 79: (22) %e A340603 22: (5,1) 53: (16) 82: (13,1) %e A340603 25: (3,3) 55: (5,3) 85: (7,3) %e A340603 27: (2,2,2) 60: (3,2,1,1) 88: (5,1,1,1) %e A340603 28: (4,1,1) 61: (18) 89: (24) %e A340603 29: (10) 62: (11,1) 90: (3,2,2,1) %t A340603 Select[Range[100],OddQ[PrimePi[FactorInteger[#][[-1,1]]]-PrimeOmega[#]]&] %Y A340603 Note: Heinz numbers are given in parentheses below. %Y A340603 These partitions are counted by A340692. %Y A340603 The complement is A340602, counted by A340601. %Y A340603 The case of positive rank is A340604. %Y A340603 - Rank - %Y A340603 A001222 gives number of prime indices. %Y A340603 A047993 counts partitions of rank 0 (A106529). %Y A340603 A061395 gives maximum prime index. %Y A340603 A101198 counts partitions of rank 1 (A325233). %Y A340603 A101707 counts partitions of odd positive rank (A340604). %Y A340603 A101708 counts partitions of even positive rank (A340605). %Y A340603 A257541 gives the rank of the partition with Heinz number n. %Y A340603 A340653 counts balanced factorizations. %Y A340603 - Odd - %Y A340603 A000009 counts partitions into odd parts (A066208). %Y A340603 A027193 counts partitions of odd length (A026424). %Y A340603 A027193 (also) counts partitions of odd maximum (A244991). %Y A340603 A058695 counts partitions of odd numbers (A300063). %Y A340603 A067659 counts strict partitions of odd length (A030059). %Y A340603 A160786 counts odd-length partitions of odd numbers (A300272). %Y A340603 A339890 counts factorizations of odd length. %Y A340603 A340102 counts odd-length factorizations into odd factors. %Y A340603 A340385 counts partitions of odd length and maximum (A340386). %Y A340603 Cf. A001221, A006141, A056239, A112798, A168659, A200750, A316413, A325134, A340608, A340609, A340610. %K A340603 nonn %O A340603 1,1 %A A340603 _Gus Wiseman_, Jan 21 2021