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

1,1

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).

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Index to sequences related to prime signature.

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... .

Example

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

Mathematica

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

Prog

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

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.

Sequence in context: A338539 A347960 A286708 * A363216 A363169 A114127

Adjacent sequences: A355459 A355460 A355461 * A355463 A355464 A355465

Keyword

nonn

Author

Amiram Eldar, Jul 03 2022

Status

approved