A360614 Numerator of the average distance between consecutive 0-prepended prime indices of n; a(1) = 0.

0, 1, 2, 1, 3, 1, 4, 1, 1, 3, 5, 2, 6, 2, 3, 1, 7, 2, 8, 1, 2, 5, 9, 1, 3, 3, 2, 4, 10, 1, 11, 1, 5, 7, 2, 1, 12, 4, 3, 3, 13, 4, 14, 5, 1, 9, 15, 2, 2, 1, 7, 2, 16, 1, 5, 1, 4, 5, 17, 3, 18, 11, 4, 1, 3, 5, 19, 7, 9, 4, 20, 2, 21, 6, 1, 8, 5, 2, 22, 3, 1, 13, 23, 1, 7, 7, 5, 5, 24, 3, 3, 3, 11, 15, 4, 1, 25, 4, 5, 3

Offset

1,3

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

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences related to prime indices in the factorization of n

Formula

Numerator of A061395(n)/A001222(n).

a(1) = 0; and for n >= 1, a(n) = A061395(n) / A366785(n) = A061395(n) / gcd(A001222(n), A061395(n)). - Antti Karttunen, Oct 23 2023

Example

The 0-prepended prime indices of 100 are {0,1,1,3,3}, with differences (1,0,2,0), with mean 3/4, so a(100) = 3.

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[If[n==1, 0, Numerator[Mean[Differences[Prepend[prix[n], 0]]]]], {n, 100}]

Prog

(PARI) A360614(n) = if(1==n, 0, my(u=primepi(vecmax(factor(n)[, 1]))); (u/gcd(u, bigomega(n)))); \\ Antti Karttunen, Oct 23 2023

Crossrefs

Positions of 1's are A340609, a superset of A106529.

For twice median instead of mean we have A360555.

The denominator is A360615.

A112798 lists prime indices, length A001222, sum A056239, max A061395.

A124010 gives prime signature, mean A088529/A088530.

A316413 lists numbers with integer mean prime index, complement A348551.

A326567/A326568 gives mean of prime indices.

Cf. A334201, A340610, A360008, A360556, A360557, A360688, A366785.

Sequence in context: A234809 A135591 A114897 * A210954 A195843 A195842

Adjacent sequences: A360611 A360612 A360613 * A360615 A360616 A360617

Keyword

nonn,frac

Author

Gus Wiseman, Feb 19 2023

Extensions

Data section extended up to a(100) by Antti Karttunen, Oct 23 2023

Status

approved