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A365174
The sum of divisors d of n such that gcd(d, n/d) is an exponentially odd number (A268335).
3
1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 27, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 51, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 108, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 107, 84, 144
OFFSET
1,2
COMMENTS
The number of these divisors is A365173(n).
LINKS
FORMULA
Multiplicative with 1 + p^e + (p^(e + 1) - p)/(p^2 - 1) if e is even, and 1 + p^e + (1 + p^(2*floor(e/4)+1))*(p^(2*floor((e+1)/4)+1) - p)/(p^2 - 1) if e is odd.
a(n) <= A000203(n), with equality if and only if n is not a biquadrateful number (A046101).
a(n) >= A034448(n), with equality if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)*zeta(6)/(2*zeta(3)) * Product_{p prime} (1 + 1/p^3 - 1/p^6) = 0.809912096042... .
MATHEMATICA
f[p_, e_] := 1 + p^e + If[EvenQ[e], (p^(e + 1) - p)/(p^2 - 1), (1 + p^(2*Floor[e/4] + 1))*(p^(2*Floor[(e + 1)/4] + 1) - p)/(p^2 - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; 1 + p^e + if(e%2, (1 + p^(2*(e\4) + 1))*(p^(2*((e+1)\4) + 1) - p)/(p^2 - 1), (p^(e+1)-p)/(p^2-1))); }
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Aug 25 2023
STATUS
approved