cp's OEIS Frontend

A357848 Heinz numbers of integer partitions whose length is twice their alternating sum.

%I A357848 #5 Oct 17 2022 07:06:57
%S A357848 1,6,15,35,40,77,84,90,143,189,210,220,221,224,250,323,364,437,462,
%T A357848 490,495,504,525,528,667,748,819,858,899,988,1029,1040,1134,1147,1155,
%U A357848 1188,1210,1320,1326,1375,1400,1408,1517,1564,1683,1690,1763,1904,1938,2021
%N A357848 Heinz numbers of integer partitions whose length is twice their alternating sum.
%C A357848 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.
%C A357848 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%e A357848 The terms together with their prime indices begin:
%e A357848      1: {}
%e A357848      6: {1,2}
%e A357848     15: {2,3}
%e A357848     35: {3,4}
%e A357848     40: {1,1,1,3}
%e A357848     77: {4,5}
%e A357848     84: {1,1,2,4}
%e A357848     90: {1,2,2,3}
%e A357848    143: {5,6}
%e A357848    189: {2,2,2,4}
%e A357848    210: {1,2,3,4}
%e A357848    220: {1,1,3,5}
%e A357848    221: {6,7}
%e A357848    224: {1,1,1,1,1,4}
%t A357848 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A357848 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A357848 Select[Range[1000],Length[primeMS[#]]==2sats[primeMS[#]]&]
%Y A357848 These partitions are counted by A357709.
%Y A357848 The version for compositions is counted by A357847.
%Y A357848 A000041 counts partitions, strict A000009.
%Y A357848 A003963 multiplies prime indices.
%Y A357848 A025047 counts alternating compositions.
%Y A357848 A056239 adds up prime indices.
%Y A357848 A103919 counts partitions by alternating sum, full triangle A344651.
%Y A357848 A357136 counts compositions by alternating sum, full triangle A097805.
%Y A357848 A357182 counts compositions w/ length = alternating sum, ranked by A357184.
%Y A357848 A357189 counts partitions w/ length = alternating sum, ranked by A357486.
%Y A357848 Cf. A000720, A001221, A001222, A262977, A301987, A357183, A357485, A357488.
%K A357848 nonn
%O A357848 1,2
%A A357848 _Gus Wiseman_, Oct 16 2022