A348617 - OEIS

%I #14 Dec 10 2021 11:11:50

%S 1,10,39,88,115,228,259,306,517,544,620,783,793,870,1150,1204,1241,

%T 1392,1656,1691,1722,1845,2369,2590,2596,2775,2944,3038,3277,3280,

%U 3339,3498,3692,3996,4247,4440,4935,5022,5170,5226,5587,5644,5875,5936,6200,6321

%N Numbers whose sum of prime indices is twice their negated alternating sum.

%C 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 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

%C 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 negated alternating sum.

%F A056239(a(n)) = -2*A316524(a(n)).

%F A346698(a(n)) = 3*A346697(a(n)).

%e The terms and their prime indices begin:

%e      1: ()

%e     10: (3,1)

%e     39: (6,2)

%e     88: (5,1,1,1)

%e    115: (9,3)

%e    228: (8,2,1,1)

%e    259: (12,4)

%e    306: (7,2,2,1)

%e    517: (15,5)

%e    544: (7,1,1,1,1,1)

%e    620: (11,3,1,1)

%e    783: (10,2,2,2)

%e    793: (18,6)

%e    870: (10,3,2,1)

%e   1150: (9,3,3,1)

%e   1204: (14,4,1,1)

%e   1241: (21,7)

%e   1392: (10,2,1,1,1,1)

%e   1656: (9,2,2,1,1,1)

%e   1691: (24,8)

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

%t ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];

%t Select[Range[1000],Total[primeMS[#]]==-2*ats[primeMS[#]]&]

%Y These partitions are counted by A001523 up to 0's.

%Y An ordered version is A349154, nonnegative A348614, reverse A349155.

%Y The nonnegative version is A349159, counted by A000712 up to 0's.

%Y The reverse nonnegative version is A349160, counted by A006330 up to 0's.

%Y A027193 counts partitions with rev-alt sum > 0, ranked by A026424.

%Y A034871, A097805, A345197 count compositions by alternating sum.

%Y A035363 = partitions with alt sum 0, ranked by A066207, complement A086543.

%Y A056239 adds up prime indices, row sums of A112798, row lengths A001222.

%Y A103919 counts partitions by alternating sum, reverse A344612.

%Y A344607 counts partitions with rev-alt sum >= 0, ranked by A344609.

%Y A346697 adds up odd-indexed prime indices.

%Y A346698 adds up even-indexed prime indices.

%Y Cf. A000984, A001700, A028260, A045931, A120452, A195017, A257991, A257992, A262977, A325698, A344619, A345958.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 26 2021