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

1, 10, 12, 39, 63, 66, 88, 112, 115, 190, 228, 255, 259, 306, 325, 408, 434, 468, 517, 544, 609, 620, 783, 793, 805, 832, 870, 931, 946, 1150, 1160, 1204, 1241, 1242, 1353, 1380, 1392, 1534, 1539, 1656, 1691, 1722, 1845, 1900, 2035, 2067, 2208, 2296, 2369

Offset: 1

Author

Gus Wiseman, Nov 25 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 reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) 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 reverse-alternating sum.

Examples

			The terms and their prime indices begin:
     1: ()
    10: (3,1)
    12: (2,1,1)
    39: (6,2)
    63: (4,2,2)
    66: (5,2,1)
    88: (5,1,1,1)
   112: (4,1,1,1,1)
   115: (9,3)
   190: (8,3,1)
   228: (8,2,1,1)
   255: (7,3,2)
   259: (12,4)
   306: (7,2,2,1)
   325: (6,3,3)
   408: (7,2,1,1,1)
   434: (11,4,1)
   468: (6,2,2,1,1)
		

Crossrefs

These partitions are counted by A006330 up to 0's.

The negative reverse version is A348617.

An ordered version is A349153, non-reverse A348614.

The non-reverse version is A349159.

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

A034871, A097805, A345197 count compositions by alternating sum.

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

A103919 counts partitions by alternating sum, reverse A344612.

A325534 counts separable partitions, ranked by A335433.

A325535 counts inseparable partitions, ranked by A335448.

A346697 adds up odd-indexed prime indices.

A346698 adds up even-indexed prime indices.

Cf. A000984, A001700, A028260, A066207, A120452, A195017, A257991, A257992, A344607, A344609, A345958, A349155.

Programs

Mathematica

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

Formula

A056239(a(n)) = 2*A344616(a(n)).

A346700(a(n)) = 3*A346699(a(n)).