A375384 - OEIS

%I #29 Sep 20 2024 06:26:17

%S 300,780,2628,3240,3828,5460,13530,18528,19110,22578,25878,31878,

%T 32640,37128,49770,56280,64980,72390,73920,78210,103740,105570,115440,

%U 137550,159330,161028,277140,288420,316410,335790,370230,386760,416328,472878,541320,664128

%N Triangular numbers that are sandwiched between two squarefree semiprimes.

%C All numbers in this sequence are even.

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

%C If they exist, further differences between this sequence and A121898 are > 10^18. - _Hugo Pfoertner_, Aug 27 2024

%C 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

%F a(n) == 0 (mod 6). - _Hugo Pfoertner_, Aug 27 2024

%e 300 (24th triangular number) between 299 = 13 * 23 and 301 = 7 * 43.

%e 780 (39th triangular number) between 779 = 19 * 41 and 781 = 11 * 71.

%e 2628 (72nd triangular number) between 2627 = 37 * 71 and 2629 = 11 * 239.

%p 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

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

%o (PARI) issp(k) = my(f=factor(k)); (bigomega(f)==2) && issquarefree(f); \\ A006881

%o 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

%Y Cf. A000217, A006881.

%Y Subsequence of A121898.

%K nonn,new

%O 1,1

%A _Massimo Kofler_, Aug 13 2024