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A375384 Triangular numbers that are sandwiched between two squarefree semiprimes.

%I A375384 #29 Sep 20 2024 06:26:17
%S A375384 300,780,2628,3240,3828,5460,13530,18528,19110,22578,25878,31878,
%T A375384 32640,37128,49770,56280,64980,72390,73920,78210,103740,105570,115440,
%U A375384 137550,159330,161028,277140,288420,316410,335790,370230,386760,416328,472878,541320,664128
%N A375384 Triangular numbers that are sandwiched between two squarefree semiprimes.
%C A375384 All numbers in this sequence are even.
%C A375384 Terms such as 120 and 528 are in A121898 but are not in this sequence.
%C A375384 If they exist, further differences between this sequence and A121898 are > 10^18. - _Hugo Pfoertner_, Aug 27 2024
%C A375384 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 A375384 a(n) == 0 (mod 6). - _Hugo Pfoertner_, Aug 27 2024
%e A375384 300 (24th triangular number) between 299 = 13 * 23 and 301 = 7 * 43.
%e A375384 780 (39th triangular number) between 779 = 19 * 41 and 781 = 11 * 71.
%e A375384 2628 (72nd triangular number) between 2627 = 37 * 71 and 2629 = 11 * 239.
%p A375384 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 A375384 q[n_] := FactorInteger[n][[;; , 2]] == {1, 1}; Select[Accumulate[Range[1100]], And @@ q /@ (# + {-1, 1}) &] (* _Amiram Eldar_, Aug 13 2024 *)
%o A375384 (PARI) issp(k) = my(f=factor(k)); (bigomega(f)==2) && issquarefree(f); \\ A006881
%o A375384 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 A375384 Cf. A000217, A006881.
%Y A375384 Subsequence of A121898.
%K A375384 nonn
%O A375384 1,1
%A A375384 _Massimo Kofler_, Aug 13 2024