A363175 -id:A363175 - cp's OEIS frontend

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

Amiram Eldar, May 19 2023

nonn

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, ... .

Amiram Eldar, Table of n, a(n) for n = 1..2151 (terms below 10^18)
Wikipedia, Powerful number.
Wikipedia, Primitive abundant number.

Intersection of A001694 and A091191.
A363175 is a subsequence.
Subsequence of A363169.
Cf. A036966.

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]

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)); }

A363177 Primitive abundant numbers (A071395) that are cubefull numbers (A036966).

Original entry on oeis.org

26376098024367, 33912126031329, 1910383099764867, 2792098376579421, 5229860083034911875, 6886512413632368153, 8815747507513708671, 28966027524687899919, 42200802302982406288, 89594138836162749375, 224439112362213402759, 288564573037131517833, 512767531125033485625

Offset: 1

Amiram Eldar, May 19 2023

nonn

It seems that this sequence is also the intersection of A036966 and A091191 (checked up to 10^27).

Are there terms that are 4-full numbers (A036967)? There are none below 10^27.

Amiram Eldar, Table of n, a(n) for n = 1..42 (terms below 10^27)
Wikipedia, Primitive abundant number.

Intersection of A036966 and A071395.
Subsequence of A363169 and A363175.
A306797 is a subsequence.
Cf. A036967, A091191.

cp's OEIS Frontend

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

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