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 A383156 #10 Apr 20 2025 02:39:02
%S A383156 0,1,1,3,1,3,1,6,3,3,1,7,1,3,3,10,1,7,1,7,3,3,1,13,3,3,6,7,1,7,1,15,3,
%T A383156 3,3,13,1,3,3,13,1,7,1,7,7,3,1,21,3,7,3,7,1,13,3,13,3,3,1,15,1,3,7,21,
%U A383156 3,7,1,7,3,7,1,22,1,3,7,7,3,7,1,21,10,3,1
%N A383156 The sum of the maximum exponents in the prime factorizations of the divisors of n.
%C A383156 Inverse Möbius transform of A051903.
%C A383156 a(n) depends only on the prime signature of n (A118914).
%H A383156 Amiram Eldar, <a href="/A383156/b383156.txt">Table of n, a(n) for n = 1..10000</a>
%H A383156 Amiram Eldar, <a href="/A383156/a383156.jpg">Plot of Sum_{k=1..n} a(k)/(c_1*n*log(n) + c_2*n) - 1 for n = 10^(1..10)</a>.
%F A383156 a(n) = Sum_{d|n} A051903(d).
%F A383156 a(n) = A000005(n) * A383157(n)/A383158(n).
%F A383156 a(p^e) = e*(e+1)/2 for prime p and e >= 0. In particular, a(p) = 1 for prime p.
%F A383156 a(n) = 2^omega(n) - 1 = A309307(n) if n is squarefree (A005117).
%F A383156 a(n) = tau(n, 2) - 1 + Sum_{k=2..A051903(n)} k * (tau(n, k+1) - tau(n, k)), where tau(n, k) is the number of k-free divisors of n (k-free numbers are numbers that are not divisible by a k-th power other than 1). For a given k >= 2, tau(n, k) is a multiplicative function with tau(p^e, k) = min(e+1, k). E.g., tau(n, 2) = A034444(n), tau(n, 3) = A073184(n), and tau(n, 4) = A252505(n).
%F A383156 Sum_{k=1..n} a(k) ~ c_1 * n * log(n) + c_2 * n, where c_1 is Niven's constant (A033150), c_2 = -1 + (2*gamma - 1)*c_1 - 2*zeta'(2)/zeta(2)^2 - Sum_{k>=3} (k-1) * (k*zeta'(k)/zeta(k)^2 - (k-1)*zeta'(k-1)/zeta(k-1)^2) = -2.37613633493572231366..., and gamma is Euler's constant (A001620).
%e A383156 4 has 3 divisors: 1, 2 = 2^1 and 4 = 2^2. The maximum exponents in their prime factorizations are 0, 1 and 2, respectively. Therefore, a(4) = 0 + 1 + 2 = 3.
%e A383156 12 has 6 divisors: 1, 2 = 2^1, 3 = 3^1, 4 = 2^2, 6 = 2 * 3 and 12 = 2^2 * 3. The maximum exponents in their prime factorizations are 0, 1, 1, 2, 1 and 2, respectively. Therefore, a(12) = 0 + 1 + 1 + 2 + 1 + 2 = 7.
%t A383156 emax[n_] := If[n == 1, 0, Max[FactorInteger[n][[;; , 2]]]]; a[n_] := DivisorSum[n, emax[#] &]; Array[a, 100]
%t A383156 (* second program: *)
%t A383156 a[n_] := If[n == 1, 0, Module[{e = FactorInteger[n][[;; , 2]], emax, v}, emax = Max[e]; v = Table[Times @@ (Min[# + 1, k + 1] & /@ e), {k, 1, emax}]; v[[1]] + Sum[k*(v[[k]] - v[[k - 1]]), {k, 2, emax}] - 1]]; Array[a, 100]
%o A383156 (PARI) emax(n) = if(n == 1, 0, vecmax(factor(n)[,2]));
%o A383156 a(n) = sumdiv(n, d, emax(d));
%o A383156 (PARI) a(n) = if(n == 1, 0, my(e = factor(n)[, 2], emax = vecmax(e), v); v = vector(emax, k, vecprod(apply(x ->min(x+1 , k+1), e))); v[1] + sum(k = 2, emax, k * (v[k]-v[k-1])) - 1);
%Y A383156 Cf. A000005, A001221, A005117, A034444, A051903, A073184, A118914, A252505, A309307, A383157, A383158, A383159.
%Y A383156 Cf. A001620, A013661, A033150, A306016.
%K A383156 nonn,easy
%O A383156 1,4
%A A383156 _Amiram Eldar_, Apr 18 2025