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

1,1

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

Links

Table of n, a(n) for n=1..36.

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A000217, A006881.

Subsequence of A121898.

Sequence in context: A250008 A332282 A190879 * A154061 A253650 A054026

Adjacent sequences: A375381 A375382 A375383 * A375385 A375386 A375387

Keyword

nonn,new

Author

Massimo Kofler, Aug 13 2024

Status

approved