A325035 Product of sums of the multisets of prime indices of each prime index of 2 * n + 1.

1, 1, 2, 2, 1, 3, 3, 2, 4, 3, 2, 4, 4, 1, 4, 5, 3, 4, 4, 3, 6, 5, 2, 5, 4, 4, 4, 6, 3, 7, 5, 2, 6, 8, 4, 5, 6, 4, 6, 6, 1, 9, 8, 4, 5, 6, 5, 6, 6, 3, 7, 6, 4, 6, 10, 4, 6, 8, 3, 8, 9, 6, 8, 11, 5, 5, 6, 2, 7, 8, 5, 9, 8, 4, 7, 6, 4, 10, 12, 4, 8, 9, 6, 8, 9, 3

Offset

0,3

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.

Links

Amiram Eldar, Table of n, a(n) for n = 0..10000

Formula

Fully multiplicative with a(prime(n)) = A056239(n), restricted to odd n.

Example

91 has prime indices {4,6} with prime indices {{1,1},{1,2}} with sums {2,3} with product a(45) = 6.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Times@@Plus@@@primeMS/@primeMS[n], {n, 1, 200, 2}]

Prog

(PARI) A056239(n) = {my(f = factor(n)); sum(i = 1, #f~, f[i, 2] * primepi(f[i, 1])); }
a(n) = {my(f = factor(2*n+1)); prod(i = 1, #f~, A056239(primepi(f[i, 1]))^f[i, 2]); } \\ Amiram Eldar, Jan 17 2026

Crossrefs

Cf. A000720, A001055, A001222, A003963, A056239, A112798, A302242, A318949.

Cf. A324926, A325031, A325032, A325033, A325034.

Sequence in context: A362500 A291564 A029266 * A348331 A035387 A242308

Adjacent sequences: A325032 A325033 A325034 * A325036 A325037 A325038

Keyword

nonn,easy

Author

Gus Wiseman, Mar 25 2019

Status

approved