A348505 a(n) = usigma(n) / gcd(sigma(n), usigma(n)), where sigma is the sum of divisors function, A000203, and usigma is the unitary sigma, A034448.

1, 1, 1, 5, 1, 1, 1, 3, 10, 1, 1, 5, 1, 1, 1, 17, 1, 10, 1, 5, 1, 1, 1, 3, 26, 1, 7, 5, 1, 1, 1, 11, 1, 1, 1, 50, 1, 1, 1, 3, 1, 1, 1, 5, 10, 1, 1, 17, 50, 26, 1, 5, 1, 7, 1, 3, 1, 1, 1, 5, 1, 1, 10, 65, 1, 1, 1, 5, 1, 1, 1, 6, 1, 1, 26, 5, 1, 1, 1, 17, 82, 1, 1, 5, 1, 1, 1, 3, 1, 10, 1, 5, 1, 1, 1, 11, 1, 50, 10, 130

Offset: 1

Author

Antti Karttunen, Oct 29 2021

Keywords

• nonn

Comments

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) = 6 != 3*10 = a(8) * a(9).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A005117, A034448, A048146, A063880, A348503, A348504, A348506 (positions of ones).

Cf. also A344697, A348049.

Programs

Mathematica

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

PARI

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A348505(n) = { my(u=A034448(n)); (u/gcd(u, sigma(n))); };

Formula

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