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A348049 a(n) = A003959(n) / gcd(sigma(n), A003959(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and sigma is the sum of divisors function.

%I A348049 #13 Oct 29 2021 09:07:19
%S A348049 1,1,1,9,1,1,1,9,16,1,1,9,1,1,1,81,1,16,1,9,1,1,1,9,36,1,8,9,1,1,1,27,
%T A348049 1,1,1,144,1,1,1,9,1,1,1,9,16,1,1,81,64,36,1,9,1,8,1,9,1,1,1,9,1,1,16,
%U A348049 729,1,1,1,9,1,1,1,144,1,1,36,9,1,1,1,81,256,1,1,9,1,1,1,9,1,16,1,9,1,1,1,27,1,64
%N A348049 a(n) = A003959(n) / gcd(sigma(n), A003959(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e and sigma is the sum of divisors function.
%C A348049 Not multiplicative. For example, a(196) = 192 != a(4) * a(49).
%H A348049 Antti Karttunen, <a href="/A348049/b348049.txt">Table of n, a(n) for n = 1..20000</a>
%H A348049 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A348049 a(n) = A003959(n) / A348047(n) = A003959(n) / gcd(A000203(n), A003959(n)).
%t A348049 f[p_, e_] := (p + 1)^e; a[1] = 1; a[n_] := (m = Times @@ f @@@ FactorInteger[n]) / GCD[m, DivisorSigma[1, n]]; Array[a, 100] (* _Amiram Eldar_, Oct 21 2021 *)
%o A348049 (PARI)
%o A348049 A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
%o A348049 A348049(n) = { my(u=A003959(n)); (u/gcd(u, sigma(n))); };
%Y A348049 Cf. A000203, A003959, A005117 (positions of 1's), A348029, A348047, A348048.
%Y A348049 Cf. also A344697.
%K A348049 nonn
%O A348049 1,4
%A A348049 _Antti Karttunen_, Oct 21 2021