A357458 First differences of A325033 = "Sum of sums of the multiset of prime indices of each prime index of n.".

0, 1, -1, 2, -1, 1, -2, 2, 0, 1, -2, 2, -1, 1, -3, 4, -2, 1, -1, 1, 0, 1, -3, 3, -1, 0, -1, 2, -1, 2, -5, 4, 0, 0, -2, 2, -1, 1, -2, 4, -3, 2, -2, 1, 0, 1, -4, 3, 0, 1, -2, 1, -1, 2, -3, 2, 0, 3, -4, 2, 0, -1, -4, 5, -1, 4, -4, 1, -1, 1, -3, 4, -2, 1, -2, 2

Offset: 1

Author

Gus Wiseman, Sep 30 2022

Keywords

• sign

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 multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

Examples

			We have A325033(5) - A325033(4) = 2 - 0, so a(4) = 2.
		

Crossrefs

The partial sums are A325033, which has row-products A325032.

The version for standard compositions is A357187.

A000961 lists prime powers.

A003963 multiples prime indices.

A005117 lists squarefree numbers.

A056239 adds up prime indices.

Cf. A000720, A001221, A001222, A007716, A109082, A275024, A302242, A302243, A302505, A324926, A325034, A357139.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Differences[Table[Plus@@Join@@primeMS/@primeMS[n],{n,100}]]

Formula

a(n) = A325033(n + 1) - A325033(n).