The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A253653 Triangular numbers that are the product of a square number and a prime number. 4
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A041785 A157848 A225674 * A267360 A046104 A116984
KEYWORD
nonn
AUTHOR
Antonio Roldán, Jan 07 2015
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 23 04:41 EDT 2024. Contains 372758 sequences. (Running on oeis4.)