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 A340929 #7 Jan 30 2021 22:51:47 %S A340929 4,12,16,18,27,40,48,60,64,72,90,100,108,112,135,150,160,162,168,192, %T A340929 225,240,243,250,252,256,280,288,352,360,375,378,392,400,420,432,448, %U A340929 528,540,567,588,600,625,630,640,648,672,700,768,792,810,832,880,882 %N A340929 Heinz numbers of integer partitions of odd negative rank. %C A340929 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions. %C A340929 The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined. %H A340929 Freeman J. Dyson, <a href="https://doi.org/10.1016/S0021-9800(69)80006-2">A new symmetry of partitions</a>, Journal of Combinatorial Theory 7.1 (1969): 56-61. %H A340929 FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000145">St000145: The Dyson rank of a partition</a> %F A340929 For all terms, A061395(a(n)) - A001222(a(n)) is odd and negative. %e A340929 The sequence of partitions together with their Heinz numbers begins: %e A340929 4: (1,1) 150: (3,3,2,1) %e A340929 12: (2,1,1) 160: (3,1,1,1,1,1) %e A340929 16: (1,1,1,1) 162: (2,2,2,2,1) %e A340929 18: (2,2,1) 168: (4,2,1,1,1) %e A340929 27: (2,2,2) 192: (2,1,1,1,1,1,1) %e A340929 40: (3,1,1,1) 225: (3,3,2,2) %e A340929 48: (2,1,1,1,1) 240: (3,2,1,1,1,1) %e A340929 60: (3,2,1,1) 243: (2,2,2,2,2) %e A340929 64: (1,1,1,1,1,1) 250: (3,3,3,1) %e A340929 72: (2,2,1,1,1) 252: (4,2,2,1,1) %e A340929 90: (3,2,2,1) 256: (1,1,1,1,1,1,1,1) %e A340929 100: (3,3,1,1) 280: (4,3,1,1,1) %e A340929 108: (2,2,2,1,1) 288: (2,2,1,1,1,1,1) %e A340929 112: (4,1,1,1,1) 352: (5,1,1,1,1,1) %e A340929 135: (3,2,2,2) 360: (3,2,2,1,1,1) %t A340929 rk[n_]:=PrimePi[FactorInteger[n][[-1,1]]]-PrimeOmega[n]; %t A340929 Select[Range[2,100],OddQ[rk[#]]&&rk[#]<0&] %Y A340929 Note: A-numbers of Heinz-number sequences are in parentheses below. %Y A340929 These partitions are counted by A101707. %Y A340929 The positive version is A101707 (A340604). %Y A340929 The even version is A101708 (A340930). %Y A340929 The not necessarily odd version is A064173 (A340788). %Y A340929 A001222 counts prime factors. %Y A340929 A027193 counts partitions of odd length (A026424). %Y A340929 A047993 counts balanced partitions (A106529). %Y A340929 A058695 counts partitions of odd numbers (A300063). %Y A340929 A061395 selects the maximum prime index. %Y A340929 A063995/A105806 count partitions by Dyson rank. %Y A340929 A072233 counts partitions by sum and length. %Y A340929 A112798 lists the prime indices of each positive integer. %Y A340929 A168659 counts partitions whose length is divisible by maximum. %Y A340929 A200750 counts partitions whose length and maximum are relatively prime. %Y A340929 - Rank - %Y A340929 A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521). %Y A340929 A101198 counts partitions of rank 1 (A325233). %Y A340929 A257541 gives the rank of the partition with Heinz number n. %Y A340929 A324516 counts partitions with rank equal to maximum minus minimum part (A324515). %Y A340929 A324518 counts partitions with rank equal to greatest part (A324517). %Y A340929 A324520 counts partitions with rank equal to least part (A324519). %Y A340929 A340601 counts partitions of even rank (A340602). %Y A340929 A340692 counts partitions of odd rank (A340603). %Y A340929 Cf. A003114, A056239, A096401, A117193, A117409, A325134, A326845, A340604, A340605, A340787, A340854/A340855. %K A340929 nonn %O A340929 1,1 %A A340929 _Gus Wiseman_, Jan 29 2021