A365174 The sum of divisors d of n such that gcd(d, n/d) is an exponentially odd number (A268335).

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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))); }

Crossrefs

Cf. A000203, A033634, A034448, A268335, A365172, A365173.

Cf. A002117, A013661, A013664.

Sequence in context: A365682 A074847 A365170 * A325317 A325316 A227131

Adjacent sequences: A365171 A365172 A365173 * A365175 A365176 A365177

Keyword

nonn,easy,mult

Author

Amiram Eldar, Aug 25 2023

Status

approved