A360615 Denominator of the average distance between consecutive 0-prepended prime indices of n; a(1) = 0.

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

Offset

1,4

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

Winston de Greef, Table of n, a(n) for n = 1..10000

Formula

Denominator of A061395(n)/A001222(n), for n>1.

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) = 4.

Mathematica

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

Prog

(PARI) a(n) = if (n==1, 0, my(f=factor(n)); denominator(primepi(vecmax(f[, 1]))/ bigomega(f))); \\ Michel Marcus, Feb 20 2023

Crossrefs

Positions of 1's are A340610

The numerator is A360614.

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. A106529, A334201, A340609, A360008, A360555, A360556, A360558.

Sequence in context: A316557 A353381 A032436 * A280274 A073408 A120454

Adjacent sequences: A360612 A360613 A360614 * A360616 A360617 A360618

Keyword

nonn,frac

Author

Gus Wiseman, Feb 19 2023

Status

approved