A348504 - OEIS

%I #12 Feb 10 2022 19:13:33

%S 1,1,1,7,1,1,1,5,13,1,1,7,1,1,1,31,1,13,1,7,1,1,1,5,31,1,10,7,1,1,1,

%T 21,1,1,1,91,1,1,1,5,1,1,1,7,13,1,1,31,57,31,1,7,1,10,1,5,1,1,1,7,1,1,

%U 13,127,1,1,1,7,1,1,1,13,1,1,31,7,1,1,1,31,121,1,1,7,1,1,1,5,1,13,1,7,1,1,1,21,1,57

%N a(n) = sigma(n) / gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

%C This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 72 = 8*9, where a(72) = 13 != 5*13 = a(8) * a(9).

%H Antti Karttunen, <a href="/A348504/b348504.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>

%F a(n) = A000203(n) / A348503(n) = A000203(n) / gcd(A000203(n), A034448(n)).

%t f1[p_, e_] := p^e + 1; f2[p_, e_] := (p^(e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := (sigma = Times @@ f2 @@@ (fct = FactorInteger[n])) / GCD[sigma, Times @@ f1 @@@ fct]; Array[a, 100] (* _Amiram Eldar_, Oct 29 2021 *)

%o (PARI)

%o A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448

%o A348504(n) = { my(u=sigma(n)); (u/gcd(u, A034448(n))); };

%Y Cf. A000203, A005117 (positions of ones), A034448, A048146, A348503, A348505.

%Y Differs from A344696 for the first time at n=72, where a(72) = 13, while A344696(72) = 65. Cf. also A348048.

%K nonn

%O 1,4

%A _Antti Karttunen_, Oct 29 2021