A360614 - OEIS

%I #16 Oct 23 2023 11:47:28

%S 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,

%T 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,

%U 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

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

%C 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.

%H Antti Karttunen, <a href="/A360614/b360614.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>

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

%F 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

%e 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.

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

%t Table[If[n==1,0,Numerator[Mean[Differences[Prepend[prix[n],0]]]]],{n,100}]

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

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

%Y For twice median instead of mean we have A360555.

%Y The denominator is A360615.

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

%Y A124010 gives prime signature, mean A088529/A088530.

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

%Y A326567/A326568 gives mean of prime indices.

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

%K nonn,frac

%O 1,3

%A _Gus Wiseman_, Feb 19 2023

%E Data section extended up to a(100) by _Antti Karttunen_, Oct 23 2023