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 A340602 #11 Apr 08 2021 03:24:33 %S A340602 1,2,5,6,8,9,11,14,17,20,21,23,24,26,30,31,32,35,36,38,39,41,44,45,47, %T A340602 49,50,54,56,57,58,59,65,66,67,68,73,74,75,80,81,83,84,86,87,91,92,95, %U A340602 96,97,99,102,103,104,106,109,110,111,120,122,124,125,126,127 %N A340602 Heinz numbers of integer partitions of even rank. %C A340602 The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is 0. %C A340602 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 A340602 FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000145">St000145: The Dyson rank of a partition</a> %F A340602 Either n = 1 or A061395(n) - A001222(n) is even. %e A340602 The sequence of partitions with their Heinz numbers begins: %e A340602 1: () 31: (11) 58: (10,1) %e A340602 2: (1) 32: (1,1,1,1,1) 59: (17) %e A340602 5: (3) 35: (4,3) 65: (6,3) %e A340602 6: (2,1) 36: (2,2,1,1) 66: (5,2,1) %e A340602 8: (1,1,1) 38: (8,1) 67: (19) %e A340602 9: (2,2) 39: (6,2) 68: (7,1,1) %e A340602 11: (5) 41: (13) 73: (21) %e A340602 14: (4,1) 44: (5,1,1) 74: (12,1) %e A340602 17: (7) 45: (3,2,2) 75: (3,3,2) %e A340602 20: (3,1,1) 47: (15) 80: (3,1,1,1,1) %e A340602 21: (4,2) 49: (4,4) 81: (2,2,2,2) %e A340602 23: (9) 50: (3,3,1) 83: (23) %e A340602 24: (2,1,1,1) 54: (2,2,2,1) 84: (4,2,1,1) %e A340602 26: (6,1) 56: (4,1,1,1) 86: (14,1) %e A340602 30: (3,2,1) 57: (8,2) 87: (10,2) %t A340602 Select[Range[100],EvenQ[PrimePi[FactorInteger[#][[-1,1]]]-PrimeOmega[#]]&] %Y A340602 Taking only length gives A001222. %Y A340602 Taking only maximum part gives A061395. %Y A340602 These partitions are counted by A340601. %Y A340602 The complement is A340603. %Y A340602 The case of positive rank is A340605. %Y A340602 - Rank - %Y A340602 A047993 counts partitions of rank 0 (A106529). %Y A340602 A101198 counts partitions of rank 1 (A325233). %Y A340602 A101707 counts partitions of odd positive rank (A340604). %Y A340602 A101708 counts partitions of even positive rank (A340605). %Y A340602 A257541 gives the rank of the partition with Heinz number n. %Y A340602 A324516 counts partitions with rank = maximum minus minimum part (A324515). %Y A340602 A340653 counts factorizations of rank 0. %Y A340602 A340692 counts partitions of odd rank (A340603). %Y A340602 - Even - %Y A340602 A024430 counts set partitions of even length. %Y A340602 A027187 counts partitions of even length (A028260). %Y A340602 A027187 (also) counts partitions of even maximum (A244990). %Y A340602 A034008 counts compositions of even length. %Y A340602 A035363 counts partitions into even parts (A066207). %Y A340602 A052841 counts ordered set partitions of even length. %Y A340602 A058696 counts partitions of even numbers (A300061). %Y A340602 A067661 counts strict partitions of even length (A030229). %Y A340602 A236913 counts even-length partitions of even numbers (A340784). %Y A340602 A339846 counts factorizations of even length. %Y A340602 Cf. A000041, A006141, A056239, A072233, A112798, A168659, A325134, A326836, A326845, A340386, A340387. %K A340602 nonn %O A340602 1,2 %A A340602 _Gus Wiseman_, Jan 21 2021