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

1,4

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.

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

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 *)

Prog

(Scheme, with memoization-macro definec)
(definec (A295879 n) (cond ((= 1 n) 1) (else (* (A008578 (A067029 n)) (A295879 (A028234 n))))))
(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

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.

Sequence in context: A358780 A336736 A000688 * A322453 A327012 A351219

Adjacent sequences: A295876 A295877 A295878 * A295880 A295881 A295882

Keyword

nonn,mult,easy,changed

Author

Antti Karttunen, Nov 29 2017

Status

approved