A253651 Triangular numbers that are the product of a triangular number and a prime number.

0, 3, 6, 15, 21, 45, 66, 78, 105, 190, 210, 231, 435, 465, 630, 861, 903, 1035, 1326, 2415, 2556, 2628, 3003, 3570, 4005, 4950, 5460, 5565, 5995, 7140, 8646, 8778, 9870, 12246, 16471, 16836, 17205, 17391, 17766, 20100, 22155, 26565, 26796, 28680, 28920, 30381, 32131, 33411, 33930, 36856

Offset: 1

Author

Antonio Roldán, Jan 07 2015

Keywords

• nonn

Examples

			190 is in the sequence because it is triangular (190=19*20/2) and 190=10*19, with 10 triangular number and 19 prime number.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..259 (all terms up to and including the 10000th triangular number)

Crossrefs

Cf. A029549 (T is 2*t), A076140 (T is 3*t), A225503 (first T to be prime(n)*t).

Cf. A188630, A253650, A253652, A253653.

Programs

Maple

N:= 10^5: # to get all terms <= N
Primes:= select(isprime, [2,seq(2*k+1,k=1..N/3)]):
select(t -> issqr(1+8*t), {seq(seq(a*(a+1)/2*p, a = 2 .. floor(sqrt(2*N/p))), p = Primes)});
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(%,list)); # Robert Israel, Jan 07 2015

Mathematica

Join[{0},Module[{nn=300,trs},trs=Accumulate[Range[nn]];Select[ trs,AnyTrue[ #/trs,PrimeQ]&]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 16 2018 *)

PARI

{i=1; j=2;print1(0,", "); while(i<=10^5, k=1; p=2; c=0; while(k1, c=k); if(c>0, print1(i, ", ")); k+=p; p+=1); i+=j; j+=1)}