cp's OEIS Frontend

A349160 Numbers whose sum of prime indices is twice their reverse-alternating sum.

%I A349160 #11 Dec 10 2021 11:14:31
%S A349160 1,10,12,39,63,66,88,112,115,190,228,255,259,306,325,408,434,468,517,
%T A349160 544,609,620,783,793,805,832,870,931,946,1150,1160,1204,1241,1242,
%U A349160 1353,1380,1392,1534,1539,1656,1691,1722,1845,1900,2035,2067,2208,2296,2369
%N A349160 Numbers whose sum of prime indices is twice their reverse-alternating sum.
%C A349160 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A349160 The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
%C A349160 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are also Heinz numbers of partitions whose sum is twice their reverse-alternating sum.
%F A349160 A056239(a(n)) = 2*A344616(a(n)).
%F A349160 A346700(a(n)) = 3*A346699(a(n)).
%e A349160 The terms and their prime indices begin:
%e A349160      1: ()
%e A349160     10: (3,1)
%e A349160     12: (2,1,1)
%e A349160     39: (6,2)
%e A349160     63: (4,2,2)
%e A349160     66: (5,2,1)
%e A349160     88: (5,1,1,1)
%e A349160    112: (4,1,1,1,1)
%e A349160    115: (9,3)
%e A349160    190: (8,3,1)
%e A349160    228: (8,2,1,1)
%e A349160    255: (7,3,2)
%e A349160    259: (12,4)
%e A349160    306: (7,2,2,1)
%e A349160    325: (6,3,3)
%e A349160    408: (7,2,1,1,1)
%e A349160    434: (11,4,1)
%e A349160    468: (6,2,2,1,1)
%t A349160 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A349160 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A349160 Select[Range[1000],Total[primeMS[#]]==2*sats[primeMS[#]]&]
%Y A349160 These partitions are counted by A006330 up to 0's.
%Y A349160 The negative reverse version is A348617.
%Y A349160 An ordered version is A349153, non-reverse A348614.
%Y A349160 The non-reverse version is A349159.
%Y A349160 A027193 counts partitions with rev-alt sum > 0, ranked by A026424.
%Y A349160 A034871, A097805, A345197 count compositions by alternating sum.
%Y A349160 A056239 adds up prime indices, row sums of A112798, row lengths A001222.
%Y A349160 A103919 counts partitions by alternating sum, reverse A344612.
%Y A349160 A325534 counts separable partitions, ranked by A335433.
%Y A349160 A325535 counts inseparable partitions, ranked by A335448.
%Y A349160 A346697 adds up odd-indexed prime indices.
%Y A349160 A346698 adds up even-indexed prime indices.
%Y A349160 Cf. A000984, A001700, A028260, A066207, A120452, A195017, A257991, A257992, A344607, A344609, A345958, A349155.
%K A349160 nonn
%O A349160 1,2
%A A349160 _Gus Wiseman_, Nov 25 2021