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A351568
Sum of the divisors of the largest unitary divisor of n that is a square.
9
1, 1, 1, 7, 1, 1, 1, 1, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 1, 31, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 91, 1, 1, 1, 1, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 1, 1, 1, 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, 1, 1, 13, 1, 7, 1, 1, 1, 1, 1, 57, 13
OFFSET
1,4
COMMENTS
Obviously, all terms are odd.
FORMULA
Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if e is even and 1 otherwise.
a(n) = A000203(A350388(n)).
a(n) = A000203(n) / A351569(n).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (zeta(3)/3) * Product_{p prime} (1 + 1/p^(3/2) + 1/p^2 - 1/p^(5/2)) = 1.008259499413... . - Amiram Eldar, Nov 20 2022
Dirichlet g.f.: zeta(2*s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s-1) - 1/p^(3*s-2)). - Amiram Eldar, Sep 03 2023
MATHEMATICA
f[p_, e_] := If[EvenQ[e], (p^(e + 1) - 1)/(p - 1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 23 2022 *)
PROG
(PARI)
A350388(n) = { my(m=1, f=factor(n)); for(k=1, #f~, if(0==(f[k, 2]%2), m *= (f[k, 1]^f[k, 2]))); (m); };
A351568(n) = sigma(A350388(n));
(Python)
from math import prod
from sympy import factorint
def A351568(n): return prod(1 if e % 2 else (p**(e+1)-1)//(p-1) for p, e in factorint(n).items()) # Chai Wah Wu, Feb 24 2022
CROSSREFS
Cf. A000203, A002117, A350388, A351569, A351570, A351575 (positions of primes).
Sequence in context: A285483 A284097 A091258 * A174544 A168423 A350071
KEYWORD
nonn,mult
AUTHOR
Antti Karttunen, Feb 23 2022
STATUS
approved