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

1,1

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

Links

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

Example

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.

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 *)

Prog

(PARI) {i=1; j=2; while(i<=3*10^6, k=1; p=3; c=0; while(k<i&&c==0, if(i/k==i\k&&isprime(i/k), c=k); if(c>0, 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

Crossrefs

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

Sequence in context: A041785 A157848 A225674 * A267360 A046104 A116984

Adjacent sequences: A253650 A253651 A253652 * A253654 A253655 A253656

Keyword

nonn

Author

Antonio Roldán, Jan 07 2015

Status

approved