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

Author

Gus Wiseman, Feb 19 2023

Keywords

• nonn
• frac

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.

Examples

			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.
		

Links

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

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.

Programs

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}]

PARI

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

Formula

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