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

1,1

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.

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 A071395.

Subsequence of A363169 and A363176.

Subsequences: A306796, A306797, A363177.

Cf. A036966.

Sequence in context: A291420 A266171 A233604 * A306796 A365611 A029568

Adjacent sequences: A363172 A363173 A363174 * A363176 A363177 A363178

Keyword

nonn

Author

Amiram Eldar, May 19 2023

Status

approved