This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A385135 #9 Jun 20 2025 08:09:29
%S A385135 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,
%T A385135 42,37,48,30,72,32,53,48,54,48,72,38,60,56,78,42,96,44,72,72,72,48,
%U A385135 104,56,90,72,84,54,111,72,104,80,90,60,144,62,96,96,106,84,144
%N A385135 The sum of divisors d of n such that n/d is an exponentially odd number (A268335).
%H A385135 Amiram Eldar, <a href="/A385135/b385135.txt">Table of n, a(n) for n = 1..10000</a>
%F A385135 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.
%F A385135 Dirichlet g.f.: zeta(s-1) * zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s)).
%F A385135 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... .
%t A385135 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]
%o A385135 (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));}
%Y A385135 Cf. A013662, A033634, A268335.
%Y A385135 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).
%K A385135 nonn,easy,mult
%O A385135 1,2
%A A385135 _Amiram Eldar_, Jun 19 2025