

A336652


Sum of positive divisors of odd part of n that are divisible by every (odd) prime dividing it: a(n) = A057723(A000265(n)).


3



1, 1, 3, 1, 5, 3, 7, 1, 12, 5, 11, 3, 13, 7, 15, 1, 17, 12, 19, 5, 21, 11, 23, 3, 30, 13, 39, 7, 29, 15, 31, 1, 33, 17, 35, 12, 37, 19, 39, 5, 41, 21, 43, 11, 60, 23, 47, 3, 56, 30, 51, 13, 53, 39, 55, 7, 57, 29, 59, 15, 61, 31, 84, 1, 65, 33, 67, 17, 69, 35, 71, 12, 73, 37, 90, 19, 77, 39, 79, 5, 120, 41, 83, 21, 85, 43
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OFFSET

1,3


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537


FORMULA

Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = (p + p^2 + ... + p^e) = sigma(p^e)1.
a(n) = A057723(A000265(n)).
a(n) = A204455(n) * A336649(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/21) * Product_{p prime} (1  1/p^2 + 1/p^3) = (Pi^2/21) * A330596 = 0.3517974711... .  Amiram Eldar, Nov 12 2022


MATHEMATICA

f[2, e_] := 1; f[p_, e_] := (p^(e+1)  1)/(p1)  1; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 07 2020 *)


PROG

(PARI) A336652(n) = if(1==n, n, my(f=factor(n)); prod(i=1, #f~, if(2==f[i, 1], 1, 1+(((f[i, 1]^(1+f[i, 2]))1) / (f[i, 1]1)))));


CROSSREFS

Cf. A000265, A000593, A057723, A204455, A330596, A336649, A336651.
Sequence in context: A348363 A081432 A318060 * A136655 A060819 A318661
Adjacent sequences: A336649 A336650 A336651 * A336653 A336654 A336655


KEYWORD

nonn,mult


AUTHOR

Antti Karttunen, Jul 30 2020


STATUS

approved



