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%I #10 May 19 2023 08:02:16
%S 196,15376,342225,570375,1032256,3172468,4636684,63126063,99198099,
%T 117234117,171991125,280495504,319600125,327921075,404529741,
%U 581549787,635689593,762155163,1029447225,1148667664,1356949503,1435045924,1501500375,1558495125,1596961444,1757705625
%N Primitive abundant numbers (A091191) that are powerful numbers (A001694).
%C The least cubefull (A036966) term is a(158) = 26376098024367 = 3^6 * 7^4 * 13^3 * 19^3.
%C A363175 is a subsequence. Terms that are not in A363175: 196, 15376, 1032256, 274810802176, 1125882727038976, 72057319160283136, ... .
%H Amiram Eldar, <a href="/A363176/b363176.txt">Table of n, a(n) for n = 1..2151</a> (terms below 10^18)
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Powerful_number">Powerful number</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Primitive_abundant_number">Primitive abundant number</a>.
%t f1[p_, e_] := (p^(e + 1) - 1)/(p^(e + 1) - p^e); f2[p_, e_] := (p^(e + 1) - p)/(p^(e + 1) - 1);
%t primAbQ[n_] := (r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f <= 2;
%t seq[max_] := Module[{pow = Union[Flatten[Table[i^2*j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}]]]}, Select[Rest[pow], primAbQ]]; seq[10^10]
%o (PARI) isPrimAb(n) = {my(f = factor(n), r, p, e); r = sigma(f, -1); r > 2 && vecmax(vector(#f~, i, p = f[i, 1]; e = f[i, 2]; (p^(e + 1) - p)/(p^(e + 1) - 1))) * r <= 2; }
%o lista(lim) = {my(pow = List(), t); for(j=1, sqrtnint(lim\1, 3), for(i=1, sqrtint(lim\j^3), listput(pow, i^2*j^3))); select(x->isPrimAb(x), Set(pow)); }
%Y Intersection of A001694 and A091191.
%Y A363175 is a subsequence.
%Y Subsequence of A363169.
%Y Cf. A036966.
%K nonn
%O 1,1
%A _Amiram Eldar_, May 19 2023
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