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 A385139 #8 Jun 20 2025 08:09:47
%S A385139 1,3,4,7,6,12,8,14,13,18,12,28,14,24,24,29,18,39,20,42,32,36,24,56,31,
%T A385139 42,39,56,30,72,32,58,48,54,48,91,38,60,56,84,42,96,44,84,78,72,48,
%U A385139 116,57,93,72,98,54,117,72,112,80,90,60,168,62,96,104,116,84,144
%N A385139 The sum of divisors d of n such that n/d has exponents in its prime factorization that are all powers of 2 (A138302).
%H A385139 Amiram Eldar, <a href="/A385139/b385139.txt">Table of n, a(n) for n = 1..10000</a>
%F A385139 Multiplicative with a(p^e) = p^e + Sum_{k=0..floor(log_2(e))} p^(e-2^k).
%F A385139 a(n) <= A000203(n), with equality if and only if n is cubefree (A004709).
%F A385139 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + (1-1/p)*(Sum_{k>=1} (Sum_{j=0..floor(log_2(k))} 1/p^(k+2^j)))) = 1.62194750148969761827... .
%t A385139 f[p_, e_] := p^e + Sum[p^(e - 2^k), {k, 0, Floor[Log2[e]]}]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A385139 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2] + sum(k = 0, logint(f[i, 2], 2), f[i, 1]^(f[i, 2]-2^k)));}
%Y A385139 Cf. A004709, A138302, A353900.
%Y A385139 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), A385135 (exponentially odd), A385136 (cubefull), A385137 (3-smooth), A385138 (5-rough), this sequence (exponentially 2^n).
%K A385139 nonn,mult,easy
%O A385139 1,2
%A A385139 _Amiram Eldar_, Jun 19 2025