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

Author

Amiram Eldar, Jun 19 2025

Keywords

• nonn
• easy
• mult

Links

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

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).

Programs

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]

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));}

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... .