A349159 Numbers whose sum of prime indices is twice their alternating sum.

1, 12, 63, 66, 112, 190, 255, 325, 408, 434, 468, 609, 805, 832, 931, 946, 1160, 1242, 1353, 1380, 1534, 1539, 1900, 2035, 2067, 2208, 2296, 2387, 2414, 2736, 3055, 3108, 3154, 3330, 3417, 3509, 3913, 4185, 4340, 4503, 4646, 4650, 4664, 4864, 5185, 5684, 5863

Offset: 1

Author

Gus Wiseman, Nov 23 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 alternating sum.

Examples

			The terms and their prime indices begin:
     1: ()
    12: (2,1,1)
    63: (4,2,2)
    66: (5,2,1)
   112: (4,1,1,1,1)
   190: (8,3,1)
   255: (7,3,2)
   325: (6,3,3)
   408: (7,2,1,1,1)
   434: (11,4,1)
   468: (6,2,2,1,1)
   609: (10,4,2)
   805: (9,4,3)
   832: (6,1,1,1,1,1,1)
   931: (8,4,4)
   946: (14,5,1)
  1160: (10,3,1,1,1)
		

Crossrefs

These partitions are counted by A000712 up to 0's.

An ordered version is A348614, negative A349154.

The negative version is A348617.

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

A025047 counts alternating or wiggly compositions, complement A345192.

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

A034871, A097805, and 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.

A116406 counts compositions with alternating sum >= 0, ranked by A345913.

A138364 counts compositions with alternating sum 0, ranked by A344619.

A325534 counts separable partitions, ranked by A335433.

A325535 counts inseparable partitions, ranked by A335448.

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. A000070, A000290, A001700, A028260, A045931, A120452, A195017, A241638, A257991, A257992, A325698, A345958, A349155.

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)).

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