A380955 - OEIS

A380955

Sum of prime indices of n (with multiplicity) minus sum of distinct prime indices of n.

0, 0, 0, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0, 3, 0, 2, 0, 1, 0, 0, 0, 2, 3, 0, 4, 1, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 1, 2, 0, 0, 3, 4, 3, 0, 1, 0, 4, 0, 2, 0, 0, 0, 1, 0, 0, 2, 5, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 3, 1, 0, 0, 0, 3, 6, 0, 0, 1, 0, 0, 0

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OFFSET

1,8

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 = 1..10000

FORMULA

a(n) = A056239(n) - A066328(n).

Additive: a(m*n) = a(m) + a(n) if gcd(m,n) = 1.

Additive with a(p^e) = (e-1)*PrimePi(p). - Amiram Eldar, Jan 17 2026

EXAMPLE

The prime indices of 96 are {1,1,1,1,1,2}, with sum 7, and with distinct prime indices {1,2}, with sum 3, so a(96) = 7 - 3 = 4.

MATHEMATICA

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Table[Total[prix[n]]-Total[Union[prix[n]]], {n, 100}]

PROG

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

CROSSREFS

Positions of 0's are A005117, complement A013929.

For length instead of sum we have A046660.

Positions of 1's are A081770.

For factors instead of indices we have A280292, firsts A280286 (sorted A381075).

A multiplicative version is A290106.

Counting partitions by this statistic gives A364916.

Dominates A374248.

Positions of first appearances are A380956, sorted A380957.

For prime multiplicities instead of prime indices we have A380958.

For product instead of sum we have A380986.

A000040 lists the primes, differences A001223.

A055396 gives least prime index, greatest A061395.

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

A304038 lists distinct prime indices, sum A066328, length A001221.

Cf. A000720, A071625, A075254, A075255, A116861, A124010, A136565, A178503, A325033, A325034, A366528, A366749, A379681.

Sequence in context: A230819 A317573 A032337 * A058190 A055736 A393041

Adjacent sequences: A380952 A380953 A380954 * A380956 A380957 A380958

KEYWORD

nonn,easy

AUTHOR

Gus Wiseman, Feb 11 2025

STATUS

approved