cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A253650 Triangular numbers that are the product of a triangular number and a square number (both greater than 1).

Original entry on oeis.org

300, 1176, 3240, 7260, 14196, 25200, 29403, 41616, 64980, 97020, 139656, 195000, 228150, 265356, 353220, 461280, 592416, 749700, 936396, 1043290, 1155960, 1412040, 1708476, 2049300, 2438736, 2881200, 3381300, 3499335, 3943836, 4573800, 5276376, 6056940, 6921060, 7874496
Offset: 1

Views

Author

Antonio Roldán, Jan 07 2015

Keywords

Examples

			3240 is in the sequence because 3240 is triangular number (3240=80*81/2), and 3240=10*324=(4*5/2)*(18^2), product of triangular number 10 and square number 324.

Links

Crossrefs

Cf. A188630, A083374, A185096, A253651, A253652, A253653.

Programs

  • Mathematica
    triQ[n_] := IntegerQ@ Sqrt[8n + 1]; lst = Sort@ Flatten@ Outer[Times, Table[ n(n + 1)/2, {n, 2, 400}], Table[ n^2, {n, 2, 200}]]; Select[ lst, triQ] (* Robert G. Wilson v, Jan 13 2015 *)
  • PARI
    {i=3; j=3; while(i<=10^7, k=3; p=3; c=0; while(k1, c=k); if(c>0, print1(i, ", ")); k+=p; p+=1); i+=j; j+=1)}
  • PARI
    is(n)=if(!ispolygonal(n,3), return(0)); fordiv(core(n,1)[2], d, d>1 && ispolygonal(n/d^2,3) && n>d^2 && return(1)); 0 \\ Charles R Greathouse IV, Sep 29 2015
  • PARI
    list(lim)=my(v=List(),t,c); for(n=24,(sqrt(8*lim+1)-1)\2, t=n*(n+1)/2; c=core(n,1)[2]*core(n+1,1)[2]; if(valuation(t,2)\2 < valuation(c,2), c/=2); fordiv(c, d, if(d>1 && ispolygonal(t/d^2,3) && t>d^2, listput(v,t); break))); Vec(v) \\ Charles R Greathouse IV, Sep 29 2015

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

Original entry on oeis.org

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

Views

Author

Antonio Roldán, Jan 07 2015

Keywords

Examples

			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.

Links

Crossrefs

Cf. A029549 (T is 2*t), A076140 (T is 3*t), A225503 (first T to be prime(n)*t).
Cf. A188630, A253650, A253652, A253653.

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    {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

Views

Author

Antonio Roldán, Jan 07 2015

Keywords

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

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
Showing 1-3 of 3 results.

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