A253653 Triangular numbers that are the product of a square number and a prime number.

3, 28, 45, 153, 171, 300, 325, 496, 2556, 2628, 3321, 4753, 4851, 7381, 8128, 13203, 19900, 25200, 25425, 29161, 29403, 56953, 64980, 65341, 101025, 166753, 195625, 209628, 320400, 354061, 388521, 389403, 468028, 662976, 664128, 749700, 750925, 780625, 781875, 936396, 1063611, 1157481

Offset: 1

Author

Antonio Roldán, Jan 07 2015

Keywords

• nonn

Comments

The perfect numbers 28, 496, 8128, ... (A000396) are in the sequence, because A000396(n) = 2^(k-1)*(2^k-1) = 2^k*(2^k-1)/2 is a triangular number, and is the product of 2^(k-1) (a square when k>2) and 2^k-1 (a Mersenne prime number).

Number of terms less than 10^n: 0, 2, 7, 14, 22, 38, 68, 100, 165, 262, 420, 667, 1064, 1754, .... - Robert G. Wilson v, Jan 11 2015

This sequence is the intersection of A000217 and A229125. - Antonio Roldán, Jan 12 2015

Examples

			45 is in the sequence because it is a triangular number (45 = 9*10/2) and 45 = 9*5, with 9 a square number and 5 a prime number.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000217, A188630, A229125, A253650, A253651, A253652.

Programs

Maple

N:= 10^7: # to get all entries <= N
Tris:= {seq(x*(x+1)/2, x = 1 .. floor((sqrt(1+8*N)-1)/2))}:
Primes:= select(isprime,[2,seq(2*i+1,i=1..floor(N/8-1))]):
Tris intersect {3,seq(seq(p*y^2,y=2..floor(sqrt(N/p))),p=Primes)};
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(%, list));  # Robert Israel, Jan 14 2015

Mathematica

tri[n_] := n(n+1)/2; fQ[n_] := Block[{exp = Sort[ Last@# & /@ FactorInteger@ n]}, exp[[1]] == 1 != exp[[2]] && Union@ Mod[ Rest@ exp, 2] == {0}]; Select[ tri@ Range@ 1500, fQ] (* Robert G. Wilson v, Jan 11 2015 *)

PARI

{i=1; j=2; while(i<=3*10^6, k=1; p=3; c=0; while(k0, print1(i, ", ")); k+=p; p+=2); i+=j; j+=1)}

PARI

lista(nn) = {for (n=1, nn, if (isprime(core(t=n*(n+1)/2)), print1(t, ", ")););} \\ Michel Marcus, Jan 12 2015