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 A351242 #40 Jul 15 2025 22:48:21
%S A351242 0,1,1,8,1,35,1,64,27,133,1,280,1,351,152,512,1,945,1,1064,370,1339,1,
%T A351242 2240,125,2205,729,2808,1,4591,1,4096,1358,4921,468,7560,1,6867,2224,
%U A351242 8512,1,12221,1,10712,4104,12175,1,17920,343,16625,4940,17640,1,25515,1456,22464
%N A351242 a(n) = n^3 * Sum_{p|n, p prime} 1/p^3.
%C A351242 Dirichlet convolution of A010051(n) and n^3. - _Wesley Ivan Hurt_, Jul 15 2025
%H A351242 Seiichi Manyama, <a href="/A351242/b351242.txt">Table of n, a(n) for n = 1..10000</a>
%F A351242 a(A000040(n)) = 1.
%F A351242 G.f.: Sum_{k>=1} x^prime(k) * (1 + 4*x^prime(k) + x^(2*prime(k))) / (1 - x^prime(k))^4. - _Ilya Gutkovskiy_, Feb 05 2022
%F A351242 Dirichlet g.f. = zeta(s-3)*primezeta(s). This follows because Sum_{n>=1} a(n)/n^s = Sum_{n>=1} (n^3/n^s) Sum_{p|n} 1/p^3. Since n = p*j, rewrite the sum as Sum_{p} Sum_{j>=1} 1/(p^3*(p*j)^(s-3)) = Sum_{p} 1/p^s Sum_{j>=1} 1/j^(s-3) = zeta(s-3)*primezeta(s). The result generalizes to higher powers of p in a(n). - _Michael Shamos_, Mar 01 2023
%F A351242 Sum_{k=1..n} a(k) ~ A085964 * n^4/4. - _Vaclav Kotesovec_, Mar 03 2023
%F A351242 a(n) = Sum_{d|n} A059376(d)*A001221(n/d). - _Ridouane Oudra_, Jul 14 2025
%F A351242 From _Wesley Ivan Hurt_, Jul 15 2025: (Start)
%F A351242 a(n) = Sum_{d|n} c(d) * (n/d)^3, where c = A010051.
%F A351242 a(p^k) = p^(3*k-3) for p prime and k>=1. (End)
%e A351242 a(6) = 35; a(6) = 6^3 * Sum_{p|6, p prime} 1/p^3 = 216 * (1/2^3 + 1/3^3) = 35.
%t A351242 a[n_]:= n^3 * Sum[1/p^3,{p,Select[Divisors[n],PrimeQ]}]; Array[a,56] (* _Stefano Spezia_, Jul 14 2025 *)
%o A351242 (PARI) a(n) = my(f = factor(n)); sum(i = 1, #f~, (n/f[i,1])^3) \\ _David A. Corneth_, Jul 14 2025
%Y A351242 Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), A322078 (k=2), this sequence (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).
%Y A351242 Cf. A000040, A001221, A010051, A059376, A085964.
%K A351242 nonn,easy
%O A351242 1,4
%A A351242 _Wesley Ivan Hurt_, Feb 05 2022