A385135 The sum of divisors d of n such that n/d is an exponentially odd number (A268335).

1, 3, 4, 6, 6, 12, 8, 13, 12, 18, 12, 24, 14, 24, 24, 26, 18, 36, 20, 36, 32, 36, 24, 52, 30, 42, 37, 48, 30, 72, 32, 53, 48, 54, 48, 72, 38, 60, 56, 78, 42, 96, 44, 72, 72, 72, 48, 104, 56, 90, 72, 84, 54, 111, 72, 104, 80, 90, 60, 144, 62, 96, 96, 106, 84, 144

Offset

1,2

Links

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

Formula

Multiplicative with a(p^e) = p^e + (p^(e+1) - 1)/(p^2-1) if e is odd, and p^e + (p^(e+1) - p)/(p^2-1) if e is even.

Dirichlet g.f.: zeta(s-1) * zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4) * Product_{p prime} (1 + 1/p^2 - 1/p^4) = 1.542116283140158741... .

Mathematica

f[p_, e_] := p^e + (p^(e+1) - If[EvenQ[e], p, 1])/(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]; p^e + (p^(e + 1) - if(e%2, 1, p))/(p^2 - 1)); }

Crossrefs

Cf. A013662, A033634, A268335.

The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), this sequence (exponentially odd), A385136 (cubefull), A385137 (3-smooth), A385138 (5-rough), A385139 (exponentially 2^n).

Sequence in context: A158523 A133689 A285895 * A220345 A349217 A065967

Adjacent sequences: A385132 A385133 A385134 * A385136 A385137 A385138

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jun 19 2025

Status

approved