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.

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A172292 Triangle read by rows: T(n, k) = (2*n+1)*(2*k+1), n>=1, 1<=k<=n.

Original entry on oeis.org

9, 15, 25, 21, 35, 49, 27, 45, 63, 81, 33, 55, 77, 99, 121, 39, 65, 91, 117, 143, 169, 45, 75, 105, 135, 165, 195, 225, 51, 85, 119, 153, 187, 221, 255, 289, 57, 95, 133, 171, 209, 247, 285, 323, 361, 63, 105, 147, 189, 231, 273, 315, 357, 399, 441, 69, 115, 161
Offset: 1

Views

Author

Vincenzo Librandi, Nov 24 2010

Keywords

Comments

A number m belongs to this sequence if and only if it is odd and composite.
First column: A016945(n, n>=1), second column: A017329(n, n>=2), third column: A147587(n, n>=3). - Vincenzo Librandi, Nov 20 2012
The number of occurrences of m corresponds to the number of nontrivial factorizations of m, i.e., A072670(m-1). - Daniel Forgues, Apr 22 2014

Examples

			Triangle begins:
9;
15, 25;
21, 35,  49;
27, 45,  63,  81;
33, 55,  77,  99,  121;
39, 65,  91,  117, 143, 169;
45, 75,  105, 135, 165, 195, 225;
51, 85,  119, 153, 187, 221, 255, 289;
57, 95,  133, 171, 209, 247, 285, 323, 361;
63, 105, 147, 189, 231, 273, 315, 357, 399, 441; etc.
Number of occurrences:
  63 = 9*7 = 21*3 has two nontrivial factorizations, thus occurs twice.

Links

Crossrefs

Cf. A144562, A083487, A016945, A017329, A147587.

Programs

  • Magma
    [4*n*k + 2*n + 2*k + 1: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 20 2012
  • Mathematica
    t[n_,k_]:= 4 n*k + 2n + 2k + 1; Table[t[n, k], {n,15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 20 2012 *)

Formula

T(n, k) = A144562(n,k)*2+3 read by rows. (Was old name.)
T(n, k) = 2*A083487(n, k)+1. - Daniel Forgues, Sep 20 2011

A176545 Numbers k>0 such that 2*k^2+14*k+5 is prime.

Original entry on oeis.org

2, 9, 11, 12, 14, 17, 21, 24, 26, 39, 44, 47, 59, 77, 86, 87, 89, 101, 116, 117, 122, 126, 131, 137, 147, 152, 161, 164, 177, 179, 189, 191, 192, 201, 212, 219, 222, 231, 249, 257, 261, 264, 284, 287, 296, 306, 317, 326, 329, 336, 347, 369, 387, 389, 411, 417
Offset: 1

Views

Author

Vincenzo Librandi, Apr 20 2010

Keywords

Comments

Conjecture: If p = 2*k^2 + 14*k + 5 then 2^p - 1 is not prime.

Links

Crossrefs

Cf. A154577 (Primes of the form 2n^2 + 14n + 5), A144562(Triangle read by rows where T(m, n) = 2m*n + m + n - 1).

Programs

  • Magma
    [n: n in [1..1000]| IsPrime(2*n^2+14*n+5)];
  • Mathematica
    Select[Range[500], PrimeQ[2#^2 + 14# + 5] &] (* Harvey P. Dale, Jan 09 2011 *)

A324937 Triangle read by rows: T(n, k) = 2*n*k + n + k - 8.

Original entry on oeis.org

-4, -1, 4, 2, 9, 16, 5, 14, 23, 32, 8, 19, 30, 41, 52, 11, 24, 37, 50, 63, 76, 14, 29, 44, 59, 74, 89, 104, 17, 34, 51, 68, 85, 102, 119, 136, 20, 39, 58, 77, 96, 115, 134, 153, 172, 23, 44, 65, 86, 107, 128, 149, 170, 191, 212, 26, 49, 72, 95, 118, 141, 164, 187, 210, 233, 256
Offset: 1

Views

Author

Vincenzo Librandi, Mar 25 2019

Keywords

Examples

			Triangle begins:
  -4;
  -1, 4;
   2, 9,  16;
   5, 14, 23, 32;
   8, 19, 30, 41, 52;
  11, 24, 37, 50, 63, 76;
  14, 29, 44, 59, 74, 89,  104;
  17, 34, 51, 68, 85, 102, 119, 136;
  20, 39, 58, 77, 96, 115, 134, 153, 172;  etc.

Crossrefs

Similar sequence T(n,k) = 2*n*k+n+k-h: A144562 (h=1); A154680 (h=2); A154684 (h=3); A155724 (h=4); A155546 (h=5); A155550 (h=6); A144670 (h=7); this sequence (h=8); A155551 (h=9).

Programs

  • Magma
    [2*n*k+n+k-8: k in [1..n], n in [1..11]]; /* As triangle */ [[2*n*k+n+k-8: k in [1..n]]: n in [1.. 15]];
  • Mathematica
    t[n_, k_]:=2 n k + n + k - 8; Table[t[n, k], {n, 11}, {k, n}]//Flatten

Formula

G.f.: x*y*(9*x^3*y^2 - 4*x^2*y*(5 + 2*y) + x*(7 + 16*y) - 4)/((1 - x)^2*(1 - x*y)^3). - Stefano Spezia, Jul 29 2025
Previous Showing 31-33 of 33 results.

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