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

1,1

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

Links

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

Example

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.

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 *)

Prog

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

Crossrefs

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

Subsequence of: A004709, A048107, A060687, A067259.

Sequence in context: A187039 A360554 A325241 * A340780 A054753 A098899

Adjacent sequences: A072354 A072355 A072356 * A072358 A072359 A072360

Keyword

nonn

Author

Reinhard Zumkeller, Jul 18 2002

Status

approved