A253650 -id:A253650 - cp's OEIS frontend

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

0, 3, 6, 15, 21, 45, 66, 78, 105, 190, 210, 231, 435, 465, 630, 861, 903, 1035, 1326, 2415, 2556, 2628, 3003, 3570, 4005, 4950, 5460, 5565, 5995, 7140, 8646, 8778, 9870, 12246, 16471, 16836, 17205, 17391, 17766, 20100, 22155, 26565, 26796, 28680, 28920, 30381, 32131, 33411, 33930, 36856

Offset: 1

Antonio Roldán, Jan 07 2015

nonn

			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.

Harvey P. Dale, Table of n, a(n) for n = 1..259 (all terms up to and including the 10000th triangular number)

Cf. A029549 (T is 2*t), A076140 (T is 3*t), A225503 (first T to be prime(n)*t).

Cf. A188630, A253650, A253652, A253653.

N:= 10^5: # to get all terms <= N
Primes:= select(isprime, [2,seq(2*k+1,k=1..N/3)]):
select(t -> issqr(1+8*t), {seq(seq(a*(a+1)/2*p, a = 2 .. floor(sqrt(2*N/p))), p = Primes)});
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(%,list)); # Robert Israel, Jan 07 2015

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 *)

{i=1; j=2;print1(0,", "); while(i<=10^5, k=1; p=2; c=0; while(k1, c=k); if(c>0, print1(i, ", ")); k+=p; p+=1); i+=j; j+=1)}

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

Original entry on oeis.org

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

Antonio Roldán, Jan 07 2015

nonn

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

			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.

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

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

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

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 *)

{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)}

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

A253652 Triangular numbers that are the product of a triangular number and an oblong number.

Original entry on oeis.org

0, 6, 36, 120, 210, 300, 630, 1176, 2016, 3240, 3570, 4950, 7140, 7260, 10296, 14196, 19110, 23436, 25200, 32640, 39060, 41616, 52326, 61776, 64980, 79800, 97020, 116886, 139656, 145530, 165600, 195000, 228150, 242556, 265356, 304590, 306936, 349866, 353220, 404550, 426426, 461280

Offset: 1

Antonio Roldán, Jan 07 2015

nonn

Supersequence of A083374, because A083374(n)= n^2 * (n^2 - 1)/2 = n*(n+1)/2*n*(n-1), product of triangular number n*(n+1)/2 and oblong number n*(n-1).

			630 is in the sequence because it is a triangular number (630 = 35*36/2) and 630 = 105*6, with 105 = 14*15/2, triangular number, and 6 = 2*3, oblong number.

Cf. A002378, A000217, A188630, A083374, A253650, A253651, A253653.

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

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

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

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A253652 Triangular numbers that are the product of a triangular number and an oblong number.

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PARI