A355462 Powerful numbers divisible by exactly 2 distinct primes.

36, 72, 100, 108, 144, 196, 200, 216, 225, 288, 324, 392, 400, 432, 441, 484, 500, 576, 648, 675, 676, 784, 800, 864, 968, 972, 1000, 1089, 1125, 1152, 1156, 1225, 1296, 1323, 1352, 1372, 1444, 1521, 1568, 1600, 1728, 1936, 1944, 2000, 2025, 2116, 2304, 2312, 2500

Offset: 1

Author

Amiram Eldar, Jul 03 2022

Keywords

• nonn

Comments

First differs from A286708 at n = 25.

Number of the form p^i * q^j, where p != q are primes and i,j > 1.

Numbers k such that A001221(k) = 2 and A051904(k) >= 2.

The possible values of the number of the divisors (A000005) of terms in this sequence is any composite number that is not 8 or twice a prime (A264828 \ {1, 8}).

675 = 3^3*5^2 and 676 = 2^2*13^2 are 2 consecutive integers in this sequence. There are no other such pairs below 10^22 (the lesser members of such pairs are terms of A060355).

Examples

			36 is a term since 36 = 2^2 * 3^2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index to sequences related to prime signature.

Crossrefs

Intersection of A001694 and A007774.

Subsequence of A286708.

Subsequences: A085986, A143610, A162142, A179646, A179666, A179671, A179689, A179694, A179699, A179702, A179705, A189988, A189990, A189991, A190464, A190465, A303661.

Cf. A000005, A001221, A051904, A060355, A136141, A264828.

Programs

Mathematica

Select[Range[2500], Length[(e = FactorInteger[#][[;; , 2]])] == 2 && Min[e] > 1 &]

PARI

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

Formula

Sum_{n>=1} 1/a(n) = ((Sum_{p prime} (1/(p*(p-1))))^2 - Sum_{p prime} (1/(p^2*(p-1)^2)))/2 = 0.1583860791... .