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A353791 Multiplicative with a(p^e) = ((p-1)*q)^e, where q is the largest prime less than p, and 1 if p = 2.

%I A353791 #15 Dec 31 2022 03:52:06
%S A353791 1,1,4,1,12,4,30,1,16,12,70,4,132,30,48,1,208,16,306,12,120,70,418,4,
%T A353791 144,132,64,30,644,48,870,1,280,208,360,16,1116,306,528,12,1480,120,
%U A353791 1722,70,192,418,1978,4,900,144,832,132,2444,64,840,30,1224,644,3074,48,3540,870,480,1,1584,280,4026,208,1672,360
%N A353791 Multiplicative with a(p^e) = ((p-1)*q)^e, where q is the largest prime less than p, and 1 if p = 2.
%H A353791 Antti Karttunen, <a href="/A353791/b353791.txt">Table of n, a(n) for n = 1..10000</a>
%H A353791 Antti Karttunen, <a href="/A353791/a353791.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H A353791 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%F A353791 a(n) = A003958(n) * A064989(n).
%F A353791 a(n) = a(2*n) = a(A000265(n)).
%F A353791 Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} ((p^3-p^2)/(p^3-p*q+q)) = 0.1075035014..., where q(p) = prevprime(p) = A151799(p) if p > 2 and q(2) = 1. - _Amiram Eldar_, Dec 31 2022
%t A353791 f[p_, e_] := (If[p == 2, 1, NextPrime[p, -1]]*(p-1))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 70] (* _Amiram Eldar_, Dec 31 2022 *)
%o A353791 (PARI) A353791(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = (f[i,1]-1)*precprime(f[i,1]-1)); factorback(f); };
%Y A353791 Cf. A003958, A064989, A151799, A353792 [= a(A000203(n))], A353793 [= a(A003961(n))], A353794 [= a(sigma(A003961(n)))].
%Y A353791 Cf. also A353749.
%K A353791 nonn,mult
%O A353791 1,3
%A A353791 _Antti Karttunen_, May 11 2022