A363176 Primitive abundant numbers (A091191) that are powerful numbers (A001694).

196, 15376, 342225, 570375, 1032256, 3172468, 4636684, 63126063, 99198099, 117234117, 171991125, 280495504, 319600125, 327921075, 404529741, 581549787, 635689593, 762155163, 1029447225, 1148667664, 1356949503, 1435045924, 1501500375, 1558495125, 1596961444, 1757705625

Offset

1,1

Comments

The least cubefull (A036966) term is a(158) = 26376098024367 = 3^6 * 7^4 * 13^3 * 19^3.

A363175 is a subsequence. Terms that are not in A363175: 196, 15376, 1032256, 274810802176, 1125882727038976, 72057319160283136, ... .

Links

Amiram Eldar, Table of n, a(n) for n = 1..2151 (terms below 10^18)

Wikipedia, Powerful number.

Wikipedia, Primitive abundant number.

Mathematica

f1[p_, e_] := (p^(e + 1) - 1)/(p^(e + 1) - p^e); f2[p_, e_] := (p^(e + 1) - p)/(p^(e + 1) - 1);
primAbQ[n_] := (r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f <= 2;
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]

Prog

(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; }
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)); }

Crossrefs

Intersection of A001694 and A091191.

A363175 is a subsequence.

Subsequence of A363169.

Cf. A036966.

Sequence in context: A285858 A203540 A006362 * A035834 A114761 A076002

Adjacent sequences: A363173 A363174 A363175 * A363177 A363178 A363179

Keyword

nonn

Author

Amiram Eldar, May 19 2023

Status

approved