A363175 Primitive abundant numbers (A071395) that are powerful numbers (A001694).

342225, 570375, 3172468, 4636684, 63126063, 99198099, 117234117, 171991125, 280495504, 319600125, 327921075, 404529741, 581549787, 635689593, 762155163, 1029447225, 1148667664, 1356949503, 1435045924, 1501500375, 1558495125, 1596961444, 1757705625, 1778362047

Offset: 1

Author

Amiram Eldar, May 19 2023

Keywords

• nonn

Comments

The least cubefull (A036966) term is a(154) = A363177(1) = 26376098024367 = 3^6 * 7^4 * 13^3 * 19^3.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..2145 (terms below 10^18)
• Wikipedia, Powerful number.
• Wikipedia, Primitive abundant number.

Crossrefs

Intersection of A001694 and A071395.

Subsequence of A363169 and A363176.

Subsequences: A306796, A306797, A363177.

Cf. A036966.

Programs

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]

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)); }