cp's OEIS Frontend

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

%I A072357 #61 Feb 24 2024 01:03:23
%S A072357 12,18,20,28,44,45,50,52,60,63,68,75,76,84,90,92,98,99,116,117,124,
%T A072357 126,132,140,147,148,150,153,156,164,171,172,175,188,198,204,207,212,
%U A072357 220,228,234,236,242,244,245,260,261,268,275,276,279,284,292,294,306,308
%N A072357 Cubefree nonsquares whose factorization into a product of primes contains exactly one square.
%C A072357 Numbers n such that A001222(n) - A001221(n) = 1 and A001221(n)>1.
%C A072357 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
%C A072357 From _Salvador Cerdá_, Mar 08 2016: (Start)
%C A072357 12!+1 = 13^2 * 2834329 is in this sequence.
%C A072357 23!+1 = 47^2 * 79 * 148139754736864591 is also in this sequence. (End)
%C A072357 The asymptotic density of this sequence is (6/Pi^2) * Sum_{p prime} 1/(p*(p+1)) (A271971). - _Amiram Eldar_, Nov 09 2020
%H A072357 Robert Israel, <a href="/A072357/b072357.txt">Table of n, a(n) for n = 1..10000</a> ( 1..100 from Paolo P. Lava)
%e A072357 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.
%p A072357 N:= 1000: # to get all terms <= N
%p A072357 Primes:= select(isprime, [$2..floor(N^(1/2))]):
%p A072357 SF:= select(numtheory:-issqrfree, [$2..N/4]):
%p A072357 S:= {seq(op(map(p -> p^2*t, select(s -> igcd(s,t)=1 and s^2*t <= N, Primes))), t = SF)}:
%p A072357 sort(convert(S,list)); # _Robert Israel_, Mar 08 2016
%t A072357 Select[Range@ 308, And[PrimeNu@ # > 1, PrimeOmega@ # - PrimeNu@ # == 1] &] (* _Michael De Vlieger_, Mar 09 2016 *)
%o A072357 (PARI) isok(n) = (omega(n) > 1) && (bigomega(n) - omega(n) == 1); \\ _Michel Marcus_, Jul 16 2015
%Y A072357 Cf. A001221, A001222, A054753 (subsequence), A271971, A325981 (conjectured subsequence).
%Y A072357 Subsequence of: A004709, A048107, A060687, A067259.
%K A072357 nonn
%O A072357 1,1
%A A072357 _Reinhard Zumkeller_, Jul 18 2002