A295879 Multiplicative with a(p) = 1, a(p^e) = prime(e-1) if e > 1.

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 7, 1, 2, 2, 4, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 1, 5, 1, 1, 1, 2, 2, 1, 1, 3, 2, 1, 1, 2, 3, 2, 1, 13

Offset: 1

Author

Antti Karttunen, Nov 29 2017

Keywords

• nonn
• mult
• easy

Comments

This sequence can be used as a filter. It matches at least to the following sequences related to the counting of various non-unitary prime divisors:

For all i, j:

a(i) = a(j) => A056170(i) = A056170(j), as A056170(n) = A001222(a(n)).

a(i) = a(j) => A162641(i) = A162641(j).

a(i) = a(j) => A295659(i) = A295659(j).

a(i) = a(j) => A295662(i) = A295662(j).

a(i) = a(j) => A295883(i) = A295883(j), as A295883(n) = A007949(a(n)).

a(i) = a(j) => A295884(i) = A295884(j).

An encoding of the prime signature of A057521(n), the powerful part of n. - Peter Munn, Apr 06 2024

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A003557, A008578, A057521, A064989, A181819.

Cf. also A293515, A294875, A294895, A294897, A295878, A351944.

Differs from A000688 for the first time at n=128, where a(128) = 13, while A000688(128) = 15.

Programs

Mathematica

Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; p > 0 :> Which[p == 1, 1, e == 1, 1, True, Prime[e - 1]]] &, 128] (* Michael De Vlieger, Nov 29 2017 *)

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, 1, prime(f[i,2]-1)));} \\ Amiram Eldar, Nov 18 2022

Formula

a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product A008578(e(i)).

a(n) = A064989(A181819(n)).

a(n) = A181819(A003557(n)). - Antti Karttunen, Apr 03 2022

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^2 + Sum_{k>=1} (prime(k+1)-prime(k))/p^(k+2)) = 2.208... . - Amiram Eldar, Nov 18 2022