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

%I A253651 #32 Nov 12 2024 22:16:55
%S A253651 0,3,6,15,21,45,66,78,105,190,210,231,435,465,630,861,903,1035,1326,
%T A253651 2415,2556,2628,3003,3570,4005,4950,5460,5565,5995,7140,8646,8778,
%U A253651 9870,12246,16471,16836,17205,17391,17766,20100,22155,26565,26796,28680,28920,30381,32131,33411,33930,36856
%N A253651 Triangular numbers that are the product of a triangular number and a prime number.
%H A253651 Harvey P. Dale, <a href="/A253651/b253651.txt">Table of n, a(n) for n = 1..259</a> (all terms up to and including the 10000th triangular number)
%e A253651 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.
%p A253651 N:= 10^5: # to get all terms <= N
%p A253651 Primes:= select(isprime, [2,seq(2*k+1,k=1..N/3)]):
%p A253651 select(t -> issqr(1+8*t), {seq(seq(a*(a+1)/2*p, a = 2 .. floor(sqrt(2*N/p))), p = Primes)});
%p A253651 # if using Maple 11 or earlier, uncomment the next line
%p A253651 # sort(convert(%,list)); # _Robert Israel_, Jan 07 2015
%t A253651 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 *)
%o A253651 (PARI) {i=1; j=2;print1(0,", "); while(i<=10^5, k=1; p=2; c=0; while(k<i&&c==0, if(i/k==i\k&&isprime(i/k)&&i/k>1, c=k); if(c>0, print1(i, ", ")); k+=p; p+=1); i+=j; j+=1)}
%Y A253651 Cf. A029549 (T is 2*t), A076140 (T is 3*t), A225503 (first T to be prime(n)*t).
%Y A253651 Cf. A188630, A253650, A253652, A253653.
%K A253651 nonn
%O A253651 1,2
%A A253651 _Antonio Roldán_, Jan 07 2015