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A348504
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.
3
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, 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, 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
OFFSET
1,4
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) = 13 != 5*13 = a(8) * a(9).
FORMULA
a(n) = A000203(n) / A348503(n) = A000203(n) / gcd(A000203(n), A034448(n)).
MATHEMATICA
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 *)
PROG
(PARI)
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A348504(n) = { my(u=sigma(n)); (u/gcd(u, A034448(n))); };
CROSSREFS
Cf. A000203, A005117 (positions of ones), A034448, A048146, A348503, A348505.
Differs from A344696 for the first time at n=72, where a(72) = 13, while A344696(72) = 65. Cf. also A348048.
Sequence in context: A088840 A348048 A348985 * A344696 A336457 A271498
KEYWORD
nonn
AUTHOR
Antti Karttunen, Oct 29 2021
STATUS
approved