A348947 - OEIS

%I #10 Feb 11 2022 11:23:53

%S 1,1,1,1,1,1,1,6,1,1,1,1,1,1,1,49,1,1,1,1,1,1,1,6,1,1,23,1,1,1,1,46,1,

%T 1,1,97,1,1,1,6,1,1,1,1,1,1,1,49,1,1,1,1,1,23,1,6,1,1,1,1,1,1,1,397,1,

%U 1,1,1,1,1,1,87,1,1,1,1,1,1,1,49,169,1,1,1,1,1,1,6,1,1,1,1,1,1,1,46,1,1,1,227

%N a(n) = A348944(n) / gcd(sigma(n), A348944(n)), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

%C Numerator of ratio A348944(n) / A000203(n).

%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 36 = 2^2 * 3^2, where a(36) = 97 <> 1 = a(4)*a(9).

%H Antti Karttunen, <a href="/A348947/b348947.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) = A348944(n) / A348946(n) = A348944(n) / gcd(A000203(n), A348944(n)).

%t f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := (p + 1)^e; f3[p_, e_] := p^e + 1; a[1] = 1; a[n_] := (s = (Times @@ f2 @@@ (f = FactorInteger[n]) + Times @@ f3 @@@ f) / 2) / GCD[Times @@ f1 @@@ f, s]; Array[a, 100] (* _Amiram Eldar_, Nov 05 2021 *)

%o (PARI)

%o A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };

%o A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };

%o A348944(n) = ((1/2)*(A003959(n)+A034448(n)));

%o A348947(n) = { my(u=A348944(n)); (u/gcd(sigma(n),u)); };

%Y Cf. A000203, A003959, A034448, A348944, A348945, A348946, A348948 (denominators).

%K nonn,frac

%O 1,8

%A _Antti Karttunen_, Nov 05 2021