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Heinz numbers of integer partitions where some part is the difference of two consecutive parts.
9

%I #5 Aug 03 2023 09:04:40

%S 6,12,18,21,24,30,36,42,48,54,60,63,65,66,70,72,78,84,90,96,102,108,

%T 114,120,126,130,132,133,138,140,144,147,150,154,156,162,165,168,174,

%U 180,186,189,192,195,198,204,210,216,222,228,231,234,240,246,252,258

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

%C In other words, partitions whose parts are not disjoint from their first differences.

%C 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 The partition {3,4,5,7} with Heinz number 6545 has first differences (1,1,2) so is not in the sequence.

%e The terms together with their prime indices begin:

%e 6: {1,2}

%e 12: {1,1,2}

%e 18: {1,2,2}

%e 21: {2,4}

%e 24: {1,1,1,2}

%e 30: {1,2,3}

%e 36: {1,1,2,2}

%e 42: {1,2,4}

%e 48: {1,1,1,1,2}

%e 54: {1,2,2,2}

%e 60: {1,1,2,3}

%e 63: {2,2,4}

%e 65: {3,6}

%e 66: {1,2,5}

%e 70: {1,3,4}

%e 72: {1,1,1,2,2}

%e 78: {1,2,6}

%e 84: {1,1,2,4}

%e 90: {1,2,2,3}

%e 96: {1,1,1,1,1,2}

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],Intersection[prix[#],Differences[prix[#]]]!={}&]

%Y For all differences of pairs the complement is A364347, counted by A364345.

%Y For all differences of pairs we have A364348, counted by A363225.

%Y Subsets of {1..n} of this type are counted by A364466, complement A364463.

%Y These partitions are counted by A364467, complement A363260.

%Y The strict case is A364536, complement A364464.

%Y A050291 counts double-free subsets, complement A088808.

%Y A323092 counts double-free partitions, ranks A320340.

%Y A325325 counts partitions with distinct first differences.

%Y Cf. A002865, A025065, A093971, A108917, A196723, A229816, A236912, A237113, A237667, A320347, A326083.

%K nonn

%O 1,1

%A _Gus Wiseman_, Aug 02 2023