A348617 Numbers whose sum of prime indices is twice their negated alternating sum.

1, 10, 39, 88, 115, 228, 259, 306, 517, 544, 620, 783, 793, 870, 1150, 1204, 1241, 1392, 1656, 1691, 1722, 1845, 2369, 2590, 2596, 2775, 2944, 3038, 3277, 3280, 3339, 3498, 3692, 3996, 4247, 4440, 4935, 5022, 5170, 5226, 5587, 5644, 5875, 5936, 6200, 6321

Offset: 1

Author

Gus Wiseman, Nov 26 2021

Keywords

• nonn

Comments

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.

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

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.

Examples

			The terms and their prime indices begin:
     1: ()
    10: (3,1)
    39: (6,2)
    88: (5,1,1,1)
   115: (9,3)
   228: (8,2,1,1)
   259: (12,4)
   306: (7,2,2,1)
   517: (15,5)
   544: (7,1,1,1,1,1)
   620: (11,3,1,1)
   783: (10,2,2,2)
   793: (18,6)
   870: (10,3,2,1)
  1150: (9,3,3,1)
  1204: (14,4,1,1)
  1241: (21,7)
  1392: (10,2,1,1,1,1)
  1656: (9,2,2,1,1,1)
  1691: (24,8)
		

Crossrefs

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

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

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

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

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

A034871, A097805, A345197 count compositions by alternating sum.

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

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

A103919 counts partitions by alternating sum, reverse A344612.

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

A346697 adds up odd-indexed prime indices.

A346698 adds up even-indexed prime indices.

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

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[1000],Total[primeMS[#]]==-2*ats[primeMS[#]]&]

Formula

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

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