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A348930
a(n) = A038502(sigma(n)), where A038502 is fully multiplicative with a(3) = 1, and a(p) = p for any other prime p.
6
1, 1, 4, 7, 2, 4, 8, 5, 13, 2, 4, 28, 14, 8, 8, 31, 2, 13, 20, 14, 32, 4, 8, 20, 31, 14, 40, 56, 10, 8, 32, 7, 16, 2, 16, 91, 38, 20, 56, 10, 14, 32, 44, 28, 26, 8, 16, 124, 19, 31, 8, 98, 2, 40, 8, 40, 80, 10, 20, 56, 62, 32, 104, 127, 28, 16, 68, 14, 32, 16, 8, 65, 74, 38, 124, 140, 32, 56, 80, 62, 121, 14, 28
OFFSET
1,3
COMMENTS
Note that a(A005820(4)) = A005820(4) and a(A005820(6)) = A005820(6), i.e., the fourth and sixth 3-perfect numbers, 459818240 and 51001180160 are among the fixed points of this sequence, precisely because they are also terms of A323653. As the former factorizes as 459818240 = 256 * 5 * 7 * 19 * 37 * 73, it must follow that a(256)/256 * a(5)/5 * a(7)/7 * a(19)/19 * a(37)/37 * a(73)/73 = 1, because ratio a(n)/n is multiplicative. See also comments in A348738.
FORMULA
Multiplicative with a(p^e) = A038502(1 + p + p^2 + ... + p^e).
a(n) = A038502(A000203(n)).
For all n >= 1, A000265(a(n)) = A336457(n).
MATHEMATICA
s[n_] := n / 3^IntegerExponent[n, 3]; Table[s[DivisorSigma[1, n]], {n, 1, 100}] (* Amiram Eldar, Nov 04 2021 *)
PROG
(PARI)
A038502(n) = (n/3^valuation(n, 3));
A348930(n) = A038502(sigma(n));
KEYWORD
nonn,mult
AUTHOR
Antti Karttunen, Nov 04 2021
STATUS
approved