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 A364990 #10 Aug 15 2023 07:43:28
%S A364990 3600,25200,28224,39600,46800,61200,68400,82800,104400,111600,133200,
%T A364990 141120,147600,154800,169200,190800,212400,219600,241200,255600,
%U A364990 262800,277200,284400,298800,310464,320400,327600,349200,363600,366912,370800,385200,392400,406800
%N A364990 Coreful triperfect numbers: numbers k such that csigma(k) = 3*k, where csigma(k) is the sum of the coreful divisors of k (A057723).
%C A364990 A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947) (the term "coreful divisor" was used by Hardy and Subbarao, 1983).
%C A364990 If k is a term, then also m*k is, for any squarefree m coprime to k. Thus there are infinitely many coreful triperfect numbers, and all of them can be generated from the sequence of primitive terms, which is the subsequence of powerful terms of this sequence. This sequence is c(n) = A064549(A005820(n)), i.e., the triperfect numbers (A005820), multiplied by their squarefree kernel (A007947): 3600, 28224, 1071645696, 1651818858099200, 532098668445696, 317519577357516800, ...
%C A364990 The asymptotic density of this sequence is Sum_{i>=1} beta(c(i))/c(i), where beta(n) = (6/Pi^2) * Product_{p|n} (p/(p+1)) = 0.0000797856... . If there are only 6 triperfect numbers, then the exact value of this density is 18575679807276818039685539/(23589576231586703755916083200 * Pi^2).
%H A364990 Amiram Eldar, <a href="/A364990/b364990.txt">Table of n, a(n) for n = 1..10000</a>
%H A364990 G. E. Hardy and M. V. Subbarao, <a href="/A005934/a005934.pdf">Highly powerful numbers</a>, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)
%e A364990 3600 is in the sequence since its coreful divisors are {30, 60, 90, 120, 150, 180, 240, 300, 360, 450, 600, 720, 900, 1200, 1800, 3600}, whose sum is 10800 = 3 * 3600.
%t A364990 f[p_, e_] := (p^(e+1)-1)/(p-1)-1; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[500000], s[#] == 3*# &]
%o A364990 (PARI) s(n) = {my(f = factor(n)); prod(i = 1, #f~, sigma(f[i, 1]^f[i, 2]) - 1);}
%o A364990 is(n) = s(n) == 3*n;
%Y A364990 Cf. A005820, A007947, A057723, A064549, A307958.
%K A364990 nonn
%O A364990 1,1
%A A364990 _Amiram Eldar_, Aug 15 2023