

A253653


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


3



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
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OFFSET

1,1


COMMENTS

The perfect numbers 28, 496, 8128, ... (A000396) are in the sequence, because A000396(n) = 2^(k1)*(2^k1) = 2^k*(2^k1)/2 is a triangular number, and is the product of 2^(k1) (a square when k>2) and 2^k1 (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

Table of n, a(n) for n=1..42.


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/81))]):
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



