A375384 Triangular numbers that are sandwiched between two squarefree semiprimes.

300, 780, 2628, 3240, 3828, 5460, 13530, 18528, 19110, 22578, 25878, 31878, 32640, 37128, 49770, 56280, 64980, 72390, 73920, 78210, 103740, 105570, 115440, 137550, 159330, 161028, 277140, 288420, 316410, 335790, 370230, 386760, 416328, 472878, 541320, 664128

Offset: 1

Author

Massimo Kofler, Aug 13 2024

Keywords

• nonn

Comments

All numbers in this sequence are even.

Terms such as 120 and 528 are in A121898 but are not in this sequence.

If they exist, further differences between this sequence and A121898 are > 10^18. - Hugo Pfoertner, Aug 27 2024

If they exist, further terms of A121898 not in this sequence are > 10^7779. This is based on considering the Diophantine equations x*(x-1) = 2*(y^2-1) and x*(x-1) = 2*(y^2+1). - Robert Israel, Sep 01 2024

Examples

			300 (24th triangular number) between 299 = 13 * 23 and 301 = 7 * 43.
780 (39th triangular number) between 779 = 19 * 41 and 781 = 11 * 71.
2628 (72nd triangular number) between 2627 = 37 * 71 and 2629 = 11 * 239.
		

Crossrefs

Cf. A000217, A006881.

Subsequence of A121898.

Programs

Maple

select(t -> numtheory:-bigomega(t+1)=2 and numtheory:-bigomega(t-1)=2 and numtheory:-issqrfree(t+1) and numtheory:-issqrfree(t-1), [seq(i*(i+1)/2, i=1..2000)]); # Robert Israel, Sep 02 2024

Mathematica

q[n_] := FactorInteger[n][[;; , 2]] == {1, 1}; Select[Accumulate[Range[1100]], And @@ q /@ (# + {-1, 1}) &] (* Amiram Eldar, Aug 13 2024 *)

PARI

issp(k) = my(f=factor(k)); (bigomega(f)==2) && issquarefree(f); \\ A006881
lista(nn) = my(list=List()); for (n=2, nn, my(k=n*(n+1)/2); if (issp(k-1) && issp(k+1), listput(list, k))); Vec(list); \\ Michel Marcus, Sep 01 2024

Formula

a(n) == 0 (mod 6). - Hugo Pfoertner, Aug 27 2024