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)}

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

Views

Author

Antonio Roldán, Jan 07 2015

Keywords

Comments

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).

Examples

			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.

Crossrefs

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

Programs

  • PARI
    {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)}
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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