

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
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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 CerdÃ¡, 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



