A048395 -id:A048395 - cp's OEIS frontend

A158526 n and (1 + 2n + 2n^2) are primes.

2, 5, 7, 17, 19, 29, 47, 79, 97, 109, 137, 139, 149, 157, 167, 199, 229, 347, 349, 389, 409, 467, 479, 547, 577, 599, 709, 719, 757, 857, 929, 937, 967, 1039, 1069, 1087, 1187, 1229, 1259, 1399, 1409, 1447, 1559, 1579, 1597, 1607, 1657, 1697, 1699, 1709

Offset: 1

Zak Seidov, Mar 20 2009

Numbers n such that A048395(n) is semiprime, or A048395(n)/n is prime.

Or, primes in A027861. Also, (1+2*n+2*n^2) are in A027862. - Zak Seidov, Sep 19 2015

			A048395(2)=26=2*13, A048395(5)=305=5*61, A048395(7)=791=7*113.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A048395 (sum of consecutive nonsquares), A001358 (semiprimes).

Cf. A027861, A027862.

[p: p in PrimesUpTo(2000) | IsPrime(2*p^2+2*p+1)]; // Vincenzo Librandi, Jun 22 2014

f[n_]:=PrimeQ[n^2+(n+1)^2];lst={};Do[p=Prime[n];If[f[p],AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)

forprime(p=2,1e4,if(isprime(2*p*(p+1)+1),print1(p","))) \\ Charles R Greathouse IV, Sep 09 2009

A364361 Table read by rows. T(n, k) = Sum_{j=0..n-k} kbinomial(k, j)binomial(n - j, k).

Original entry on oeis.org

0, 0, 1, 0, 3, 2, 0, 5, 10, 3, 0, 7, 26, 21, 4, 0, 9, 50, 75, 36, 5, 0, 11, 82, 189, 164, 55, 6, 0, 13, 122, 387, 516, 305, 78, 7, 0, 15, 170, 693, 1284, 1155, 510, 105, 8, 0, 17, 226, 1131, 2724, 3405, 2262, 791, 136, 9, 0, 19, 290, 1725, 5156, 8415, 7734, 4025, 1160, 171, 10

Offset: 0

Peter Luschny, Jul 30 2023

			The triangle begins:
  [0] 0;
  [1] 0,  1;
  [2] 0,  3,   2;
  [3] 0,  5,  10,   3;
  [4] 0,  7,  26,   21,    4;
  [5] 0,  9,  50,   75,   36,    5;
  [6] 0, 11,  82,  189,  164,   55,    6;
  [7] 0, 13, 122,  387,  516,  305,   78,   7;
  [8] 0, 15, 170,  693, 1284, 1155,  510, 105,   8;
  [9] 0, 17, 226, 1131, 2724, 3405, 2262, 791, 136, 9;
Seen as an array:
  [0] 0,  1,   2,   3,     4,     5,      6,      7, ...  A001477
  [1] 0,  3,  10,   21,   36,    55,     78,    105, ...  A014105
  [2] 0,  5,  26,   75,  164,   305,    510,    791, ...  A048395
  [3] 0,  7,  50,  189,  516,  1155,   2262,   4025, ...
  [4] 0,  9,  82,  387, 1284,  3405,   7734,  15687, ...
  [5] 0, 11, 122,  693, 2724,  8415,  21918,  50281, ...
  [6] 0, 13, 170, 1131, 5156, 18265,  53934, 138775, ...
  [7] 0, 15, 226, 1725, 8964, 35915, 118950, 340473, ...
    A005408|A069894

Cf. A364553 (row sums),  A364634 (main diagonal).
Rows: A001477, A014105, A048395.
Columns: A005408, A069894.

T := (n, k) -> local j; add(k*binomial(k, j)*binomial(n-j, k), j = 0..n-k):
seq(seq(T(n, k), k = 0..n), n = 0..10);

T(2*n, n) = n * LegendreP(n, 3).

A139309 Array by antidiagonals, sum of non-k-gonal numbers between consecutive k-gonal numbers.

Original entry on oeis.org

0, 0, 2, 0, 5, 9, 0, 9, 26, 24, 0, 14, 51, 75, 50, 0, 20, 84, 153, 164, 90, 0, 27, 125, 258, 342, 305, 147, 0, 35, 174, 390, 584, 645, 510, 224, 0, 44, 231, 549, 890, 1110, 1089, 791, 324, 0, 54, 296, 735, 1260, 1700, 1884, 1701, 1160, 450, 0, 65, 369, 948, 1694

Offset: 0

Jonathan Vos Post, Jun 07 2008

The n=1 column is A000096(k) = n*(n+3)/2. The k=3 row is the sum of nontriangular numbers between successive triangular numbers (A006002) = the sum of n consecutive integers beginning with (n-th triangular number)+1 = (n*(n+1)^2)/2. The k=4 row is the sum of nonsquares between successive squares (A048395) = 2*n^3 + 2*n^2 + n. The k=5 row is the sum of non-pentagonal numbers between successive pentagonal numbers. The k-th row is the sum of non-k-gonal numbers between successive k-gonal numbers. Each column is a quadratic sequence. Each row is a cubic sequence.

			The array begins:
========================================================================
...|.n=0.|.n=1.|.n=2.|.n=3.|.n=4.|.n=5.|.n=6.|.n=7.|.n=8.|.n=9.|.in.OEIS
====|=====|=====|=====|=====|=====|=====|=====|=====|=====|=====|========
k=3.|..0..|..2..|..9..|..24.|..50.|..90.|.147.|.224.|.324.|.450.|.A006002
k=4.|..0..|..5..|.26..|..75.|.164.|.305.|.510.|.791.|1160.|1629.|.A048395
k=5.|..0..|..9..|.51..|.153.|.342.|.645.|1089.|...................not.yet
k=6.|..0..|.14..|.84..|.258.|.584.|...............................not.yet
k=7.|..0..|.20..|125..|.390.|.....................................not.yet
k=8.|..0..|.27..|174..|...........................................not.yet
k=9.|..0..|.35..|231..|...........................................not.yet
k=10|..0..|.44..|296..|...........................................not.yet
========================================================================

Cf. A000027, A000096, A006002, A048395.

A139309 := proc(k,n) n*(k-2)*((k-2)*n^2+1+2*n)/2 ; end: for d from 3 to 16 do for n from 0 to d-3 do printf("%d,", A139309(d-n,n)) ; od: od: # R. J. Mathar, Jun 12 2008

T(k,n) = n(k-2)((k-2)n^2+1+2n)/2. - R. J. Mathar, Jun 12 2008

More terms from R. J. Mathar, Jun 12 2008

cp's OEIS Frontend

A158526 n and (1 + 2n + 2n^2) are primes.

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A364361 Table read by rows. T(n, k) = Sum_{j=0..n-k} kbinomial(k, j)binomial(n - j, k).

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A139309 Array by antidiagonals, sum of non-k-gonal numbers between consecutive k-gonal numbers.

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cp's OEIS Frontend

A158526 n and (1 + 2*n + 2*n^2) are primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A364361 Table read by rows. T(n, k) = Sum_{j=0..n-k} k*binomial(k, j)*binomial(n - j, k).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula

A139309 Array by antidiagonals, sum of non-k-gonal numbers between consecutive k-gonal numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

Extensions

A158526 n and (1 + 2n + 2n^2) are primes.

A364361 Table read by rows. T(n, k) = Sum_{j=0..n-k} kbinomial(k, j)binomial(n - j, k).