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%I #31 Apr 06 2024 20:33:03
%S 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,
%T 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,
%U 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
%N Multiplicative with a(p) = 1, a(p^e) = prime(e-1) if e > 1.
%C 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:
%C For all i, j:
%C a(i) = a(j) => A056170(i) = A056170(j), as A056170(n) = A001222(a(n)).
%C a(i) = a(j) => A162641(i) = A162641(j).
%C a(i) = a(j) => A295659(i) = A295659(j).
%C a(i) = a(j) => A295662(i) = A295662(j).
%C a(i) = a(j) => A295883(i) = A295883(j), as A295883(n) = A007949(a(n)).
%C a(i) = a(j) => A295884(i) = A295884(j).
%C An encoding of the prime signature of A057521(n), the powerful part of n. - _Peter Munn_, Apr 06 2024
%H Antti Karttunen, <a href="/A295879/b295879.txt">Table of n, a(n) for n = 1..65537</a>
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product A008578(e(i)).
%F a(n) = A064989(A181819(n)).
%F a(n) = A181819(A003557(n)). - _Antti Karttunen_, Apr 03 2022
%F 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
%t 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 *)
%o (Scheme, with memoization-macro definec)
%o (definec (A295879 n) (cond ((= 1 n) 1) (else (* (A008578 (A067029 n)) (A295879 (A028234 n))))))
%o (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
%Y Cf. A003557, A008578, A057521, A064989, A181819.
%Y Cf. also A293515, A294875, A294895, A294897, A295878, A351944.
%Y Differs from A000688 for the first time at n=128, where a(128) = 13, while A000688(128) = 15.
%K nonn,mult,easy
%O 1,4
%A _Antti Karttunen_, Nov 29 2017
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