A348947 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.

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, 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, 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

Offset

1,8

Comments

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

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).

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences related to sigma(n)

Formula

a(n) = A348944(n) / A348946(n) = A348944(n) / gcd(A000203(n), A348944(n)).

Mathematica

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 *)

Prog

(PARI)
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
A348944(n) = ((1/2)*(A003959(n)+A034448(n)));
A348947(n) = { my(u=A348944(n)); (u/gcd(sigma(n), u)); };

Crossrefs

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

Sequence in context: A257936 A348039 A143532 * A267426 A202917 A324396

Adjacent sequences: A348944 A348945 A348946 * A348948 A348949 A348950

Keyword

nonn,frac

Author

Antti Karttunen, Nov 05 2021

Status

approved