A363169 Powerful abundant numbers: numbers that are both powerful (A001694) and abundant (A005101).

36, 72, 100, 108, 144, 196, 200, 216, 288, 324, 392, 400, 432, 500, 576, 648, 784, 800, 864, 900, 968, 972, 1000, 1152, 1296, 1352, 1372, 1568, 1600, 1728, 1764, 1800, 1936, 1944, 2000, 2304, 2500, 2592, 2700, 2704, 2744, 2916, 3136, 3200, 3456, 3528, 3600, 3872, 3888, 4000

Offset: 1

Author

Amiram Eldar, May 19 2023

Keywords

• nonn

Comments

The least odd term is a(90) = 11025, and the least term that is coprime to 6 is 1382511906801025.

Are there two consecutive integers in this sequence? There are none below 10^22.

Examples

			36 = 2^2 * 3^2 is a term since it is powerful, and sigma(36) = 91 > 2*36 = 72.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10534 (terms not exceeding 10^8)

Crossrefs

Intersection of A001694 and A005101.

Cf. A180114, A363170, A363171.

Subsequences: A307959, A328136, A356871.

Programs

Mathematica

Select[Range[4000], DivisorSigma[-1, #] > 2 && Min[FactorInteger[#][[;;, 2]]] > 1 &]

PARI

is(n) = { my(f = factor(n)); n > 1 && vecmin(f[, 2]) > 1 && sigma(f, -1) > 2; }