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A072357 Cubefree nonsquares whose factorization into a product of primes contains exactly one square. 4
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 (list; graph; refs; listen; history; text; internal format)
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)

LINKS

Paolo P. Lava and 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), A325981 (conjectured subsequence).

Subsequence of: A004709, A048107, A060687, A067259.

Sequence in context: A267117 A187039 A325241 * A054753 A098899 A098770

Adjacent sequences:  A072354 A072355 A072356 * A072358 A072359 A072360

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Jul 18 2002

STATUS

approved

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Last modified October 20 17:28 EDT 2019. Contains 328268 sequences. (Running on oeis4.)