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A072357 Cubefree nonsquares whose factorization into a product of primes contains exactly one square. 6

%I #61 Feb 24 2024 01:03:23

%S 12,18,20,28,44,45,50,52,60,63,68,75,76,84,90,92,98,99,116,117,124,

%T 126,132,140,147,148,150,153,156,164,171,172,175,188,198,204,207,212,

%U 220,228,234,236,242,244,245,260,261,268,275,276,279,284,292,294,306,308

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

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

%C 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 From _Salvador Cerdá_, Mar 08 2016: (Start)

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

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

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

%H Robert Israel, <a href="/A072357/b072357.txt">Table of n, a(n) for n = 1..10000</a> ( 1..100 from Paolo P. Lava)

%e 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 N:= 1000: # to get all terms <= N

%p Primes:= select(isprime, [$2..floor(N^(1/2))]):

%p SF:= select(numtheory:-issqrfree, [$2..N/4]):

%p S:= {seq(op(map(p -> p^2*t, select(s -> igcd(s,t)=1 and s^2*t <= N, Primes))), t = SF)}:

%p sort(convert(S,list)); # _Robert Israel_, Mar 08 2016

%t Select[Range@ 308, And[PrimeNu@ # > 1, PrimeOmega@ # - PrimeNu@ # == 1] &] (* _Michael De Vlieger_, Mar 09 2016 *)

%o (PARI) isok(n) = (omega(n) > 1) && (bigomega(n) - omega(n) == 1); \\ _Michel Marcus_, Jul 16 2015

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

%Y Subsequence of: A004709, A048107, A060687, A067259.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, Jul 18 2002

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Last modified March 29 08:53 EDT 2024. Contains 371268 sequences. (Running on oeis4.)