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 A337032 #17 Jan 10 2025 04:34:32
%S A337032 0,150,8400,150300,1394400,8656200,40356000,153679800,498153600,
%T A337032 1431378900,3705270000,8863150800,19694152800,41402744400,82382680800,
%U A337032 157380332400,288000115200,511088547150,875865085200,1465721632200,2382961862400,3801687211800,5918070367200,9075809181600
%N A337032 a(n) = (n*sigma_9(n) - tau(n))/7 = (A282254(n) - A000594(n))/7, where tau is Ramanujan's tau, sigma_9(n) = Sum_{d divides n} d^9.
%C A337032 D. H. Lehmer shows that tau(n) == n*sigma_9(n) (mod 7), so a(n) is an integer for all n. Furthermore, if n == 3, 5, 6 (mod 7) then tau(n) == n*sigma_9(n) (mod 49). See the Wikipedia link below. It seems that the latter congruence also holds for most of the other numbers. Among the 571 numbers in [1, 1000] congruent to 0, 1, 2, 4 modulo 7, tau(n) == n*sigma_9(n) holds for 311 n's, and among the 5715 numbers in [1, 10000] congruent to 0, 1, 2, 4 modulo 7, the congruence holds for 3492 n's.
%C A337032 It seems that 150 divides a(n) for all n. There are no counterexamples for n <= 10000.
%C A337032 Number of n's in [2, N] which satisfy the higher-order congruence tau(n) == n*sigma_9(n) (mod 7^e) but not tau(n) == n*sigma_9(n) (mod 7^(e+1)):
%C A337032 N = 1000:
%C A337032 e | n == 3, 5, 6 (mod 7) | n == 0, 1, 2, 4 (mod 7) | total
%C A337032 ---+----------------------+-------------------------+-------
%C A337032 1 | 0 | 260 | 260
%C A337032 ---+----------------------+-------------------------+-------
%C A337032 2 | 358 | 80 | 438
%C A337032 ---+----------------------+-------------------------+-------
%C A337032 3 | 45 | 195 | 240
%C A337032 ---+----------------------+-------------------------+-------
%C A337032 4 | 24 | 28 | 52
%C A337032 ---+----------------------+-------------------------+-------
%C A337032 5 | 2 | 5 | 7
%C A337032 ---+----------------------+-------------------------+-------
%C A337032 6 | 0 | 2* | 2
%C A337032 * n = 686, 942.
%C A337032 N = 10000:
%C A337032 e | n == 3, 5, 6 (mod 7) | n == 0, 1, 2, 4 (mod 7) | total
%C A337032 ---+----------------------+-------------------------+-------
%C A337032 1 | 0 | 2223 | 2223
%C A337032 ---+----------------------+-------------------------+-------
%C A337032 2 | 3368 | 728 | 4096
%C A337032 ---+----------------------+-------------------------+-------
%C A337032 3 | 466 | 2280 | 2746
%C A337032 ---+----------------------+-------------------------+-------
%C A337032 4 | 397 | 384 | 781
%C A337032 ---+----------------------+-------------------------+-------
%C A337032 5 | 46 | 87 | 133
%C A337032 ---+----------------------+-------------------------+-------
%C A337032 6 | 6 | 12 | 18
%C A337032 ---+----------------------+-------------------------+-------
%C A337032 7 | 2** | 0 | 2
%C A337032 ** n = 5185, 9021.
%H A337032 Amiram Eldar, <a href="/A337032/b337032.txt">Table of n, a(n) for n = 1..10000</a>
%H A337032 Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Ramanujan_tau_function#Congruences_for_the_tau_function">Congruences for the tau function</a>.
%e A337032 a(2) = (n*sigma_9(2) - tau(2))/7 = (2*(1^9+2^9) - (-24))/7 = 1050/7 = 150;
%e A337032 a(3) = (n*sigma_9(3) - tau(3))/7 = (3*(1^9+3^9) - 252)/7 = 58800/7 = 8400.
%t A337032 a[n_] := (n * DivisorSigma[9, n] - RamanujanTau[n]) / 7; Array[a, 24] (* _Amiram Eldar_, Jan 10 2025 *)
%o A337032 (PARI) a(n) = (n*sigma(n, 9) - polcoeff( x * eta(x + x * O(x^n))^24, n))/7;
%Y A337032 Cf. A000594, A282254, A027860.
%K A337032 nonn
%O A337032 1,2
%A A337032 _Jianing Song_, Aug 12 2020