A072357 Cubefree nonsquares whose factorization into a product of primes contains exactly one square.

12, 18, 20, 28, 44, 45, 50, 52, 60, 63, 68, 75, 76, 84, 90, 92, 98, 99, 116, 117, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 171, 172, 175, 188, 198, 204, 207, 212, 220, 228, 234, 236, 242, 244, 245, 260, 261, 268, 275, 276, 279, 284, 292, 294, 306, 308

Offset: 1

Author

Reinhard Zumkeller, Jul 18 2002

Keywords

• nonn

Comments

Numbers n such that A001222(n) - A001221(n) = 1 and A001221(n)>1.

Numbers with one or more 1's, exactly one 2 and no 3's or higher in their prime exponents. - Antti Karttunen, Sep 19 2019

From Salvador Cerdá, Mar 08 2016: (Start)

12!+1 = 13^2 * 2834329 is in this sequence.

23!+1 = 47^2 * 79 * 148139754736864591 is also in this sequence. (End)

The asymptotic density of this sequence is (6/Pi^2) * Sum_{p prime} 1/(p*(p+1)) (A271971). - Amiram Eldar, Nov 09 2020

Examples

			a(14) = 84 = 7*3*2^2; the following numbers are not terms: 36=6^2, as it is a square; 54=2*3^3, as it is not cubefree; 42=2*3*7, as there is no squared prime; 72=2*6^2, as 72 has two squared prime divisors: 2^2 and 3^2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000 ( 1..100 from Paolo P. Lava)

Crossrefs

Cf. A001221, A001222, A054753 (subsequence), A271971, A325981 (conjectured subsequence).

Subsequence of: A004709, A048107, A060687, A067259.

Programs

Maple

N:= 1000: # to get all terms <= N
Primes:= select(isprime, [$2..floor(N^(1/2))]):
SF:= select(numtheory:-issqrfree, [$2..N/4]):
S:= {seq(op(map(p -> p^2*t, select(s -> igcd(s,t)=1 and s^2*t <= N, Primes))), t = SF)}:
sort(convert(S,list)); # Robert Israel, Mar 08 2016

Mathematica

Select[Range@ 308, And[PrimeNu@ # > 1, PrimeOmega@ # - PrimeNu@ # == 1] &] (* Michael De Vlieger, Mar 09 2016 *)

PARI

isok(n) = (omega(n) > 1) && (bigomega(n) - omega(n) == 1); \\ Michel Marcus, Jul 16 2015