cp's OEIS Frontend

A364537 Heinz numbers of integer partitions where some part is the difference of two consecutive parts.

%I A364537 #5 Aug 03 2023 09:04:40
%S A364537 6,12,18,21,24,30,36,42,48,54,60,63,65,66,70,72,78,84,90,96,102,108,
%T A364537 114,120,126,130,132,133,138,140,144,147,150,154,156,162,165,168,174,
%U A364537 180,186,189,192,195,198,204,210,216,222,228,231,234,240,246,252,258
%N A364537 Heinz numbers of integer partitions where some part is the difference of two consecutive parts.
%C A364537 In other words, partitions whose parts are not disjoint from their first differences.
%C A364537 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.
%e A364537 The partition {3,4,5,7} with Heinz number 6545 has first differences (1,1,2) so is not in the sequence.
%e A364537 The terms together with their prime indices begin:
%e A364537    6: {1,2}
%e A364537   12: {1,1,2}
%e A364537   18: {1,2,2}
%e A364537   21: {2,4}
%e A364537   24: {1,1,1,2}
%e A364537   30: {1,2,3}
%e A364537   36: {1,1,2,2}
%e A364537   42: {1,2,4}
%e A364537   48: {1,1,1,1,2}
%e A364537   54: {1,2,2,2}
%e A364537   60: {1,1,2,3}
%e A364537   63: {2,2,4}
%e A364537   65: {3,6}
%e A364537   66: {1,2,5}
%e A364537   70: {1,3,4}
%e A364537   72: {1,1,1,2,2}
%e A364537   78: {1,2,6}
%e A364537   84: {1,1,2,4}
%e A364537   90: {1,2,2,3}
%e A364537   96: {1,1,1,1,1,2}
%t A364537 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A364537 Select[Range[100],Intersection[prix[#],Differences[prix[#]]]!={}&]
%Y A364537 For all differences of pairs the complement is A364347, counted by A364345.
%Y A364537 For all differences of pairs we have A364348, counted by A363225.
%Y A364537 Subsets of {1..n} of this type are counted by A364466, complement A364463.
%Y A364537 These partitions are counted by A364467, complement A363260.
%Y A364537 The strict case is A364536, complement A364464.
%Y A364537 A050291 counts double-free subsets, complement A088808.
%Y A364537 A323092 counts double-free partitions, ranks A320340.
%Y A364537 A325325 counts partitions with distinct first differences.
%Y A364537 Cf. A002865, A025065, A093971, A108917, A196723, A229816, A236912, A237113, A237667, A320347, A326083.
%K A364537 nonn
%O A364537 1,1
%A A364537 _Gus Wiseman_, Aug 02 2023