A155724 -id:A155724 - cp's OEIS frontend

A155722 Numbers k such that 2*k + 9 is prime.

1, 2, 4, 5, 7, 10, 11, 14, 16, 17, 19, 22, 25, 26, 29, 31, 32, 35, 37, 40, 44, 46, 47, 49, 50, 52, 59, 61, 64, 65, 70, 71, 74, 77, 79, 82, 85, 86, 91, 92, 94, 95, 101, 107, 109, 110, 112, 115, 116, 121, 124, 127, 130, 131, 134, 136, 137, 142, 149, 151, 152, 154, 161, 164

Offset: 1

Vincenzo Librandi, Jan 25 2009

Subsequence of A001651; A011655(a(n)) = 1. - Reinhard Zumkeller, Jul 09 2010

One less than the associated entry in A105760, two less than in A089038, three less than in A067076. - R. J. Mathar, Jan 05 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A155723, A155724.

Numbers h such that 2*h + k is prime: A005097 (k=1), A067076 (k=3), A089038 (k=5), A105760 (k=7), this seq (k=9), A101448 (k=11), A153081 (k=13), A089559 (k=15), A173059 (k=17), A153143 (k=19).

Filtered([0..200], k-> IsPrime(2*k+9)); # G. C. Greubel, May 21 2019

[n: n in [0..200] | IsPrime(2*n+9)]; // Vincenzo Librandi, Sep 24 2012

(Prime[Range[5, 100]] - 9)/2 (* Vladimir Joseph Stephan Orlovsky, Feb 08 2010 *)
Select[Range[0, 200], PrimeQ[2 # + 9]&] (* Vincenzo Librandi, Sep 24 2012 *)

is(n)=isprime(2*n+9) \\ Charles R Greathouse IV, Jul 12 2016

[n for n in (0..200) if is_prime(2*n+9) ] # G. C. Greubel, May 21 2019

Edited by N. J. A. Sloane, Jun 23 2010

Definition clarified by Zak Seidov, Jul 11 2014

A154685 Triangle read by rows: T(n, k) = 2nk + n + k + 4.

Original entry on oeis.org

8, 11, 16, 14, 21, 28, 17, 26, 35, 44, 20, 31, 42, 53, 64, 23, 36, 49, 62, 75, 88, 26, 41, 56, 71, 86, 101, 116, 29, 46, 63, 80, 97, 114, 131, 148, 32, 51, 70, 89, 108, 127, 146, 165, 184, 35, 56, 77, 98, 119, 140, 161, 182, 203, 224, 38, 61, 84, 107, 130, 153, 176, 199, 222, 245, 268

Offset: 1

Vincenzo Librandi, Jan 18 2009

The terms form a subset of A153039 because 2*T(n, k) - 7 = (2*n+1)*(2*k+1) are not prime.

			Triangle begins:
   8;
  11, 16;
  14, 21, 28;
  17, 26, 35, 44;
  20, 31, 42, 53,  64;
  23, 36, 49, 62,  75,  88;
  26, 41, 56, 71,  86, 101, 116;
  29, 46, 63, 80,  97, 114, 131, 148;
  32, 51, 70, 89, 108, 127, 146, 165, 184;
  35, 56, 77, 98, 119, 140, 161, 182, 203, 224;

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A089192, A153039.
Cf. A151675 (row sums).
Similar triangle: A155724.
Columns k: A016789 (k=1), A016861 (k=2).
Main diagonal: A137882, A271649.

A154685:= func< n,k | 2*n*k+n+k+4 >;
[A154685(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jan 21 2025

Flatten@Table[2*n*m+m+n+4,{n,20},{m,n}] (* Vincenzo Librandi, Jan 29 2012 *)

for(m=1,9,for(n=1,m,print1(2*m*n+m+n+4", "))) \\ Charles R Greathouse IV, Dec 27 2011

def A154685(n,k): return 2*n*k+n+k+4
print(flatten([[A154685(n,k) for k in range(1,n+1)] for n in range(1,16)])) # G. C. Greubel, Jan 21 2025

Sum_{k=1..n} T(n, k) = A151675(n). - N. J. A. Sloane, May 31 2009
T(n, k) = A155724(n,k) + 8. - L. Edson Jeffery, Oct 12 2012
From G. C. Greubel, Jan 21 2025: (Start)
T(2*n-1, n) = 4*n^2 + n + 3.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (1/4)*(9*(1-(-1)^n) + 2*(2-3*(-1)^n)*n - 4*(-1)^n*n^2).
G.f.: x*y*(8 - 5*(x+y) + 4*x*y)/((1-x)*(1-y))^2.
E.g.f.: 4 - (4+x)*exp(x) - (4+y)*exp(y) + (4+x+y+2*x*y)*exp(x+y).
 (End)

Clarified comment. - R. J. Mathar, Jan 24 2009

A155723 Numbers k such that 2*k + 9 is not prime.

Original entry on oeis.org

0, 3, 6, 8, 9, 12, 13, 15, 18, 20, 21, 23, 24, 27, 28, 30, 33, 34, 36, 38, 39, 41, 42, 43, 45, 48, 51, 53, 54, 55, 56, 57, 58, 60, 62, 63, 66, 67, 68, 69, 72, 73, 75, 76, 78, 80, 81, 83, 84, 87, 88, 89, 90, 93, 96, 97, 98, 99, 100, 102, 103, 104, 105, 106, 108, 111, 113, 114

Offset: 1

Vincenzo Librandi, Jan 25 2009

2*A155724(m,n) + 9 = (2n+1)*(2m+1) are not prime and create entries of this form. Also, one less than the associate entry in A153053, two less than the associated A153052. - R. J. Mathar, Jan 05 2011

			Distribution of the terms in the following triangular array:
   0;
   3,  8;
   6, 13, 20;
   9, 18, 27,  36;
  12, 23, 34,  45,  56;
  15, 28, 41,  54,  67,  80;
  18, 33, 48,  63,  78,  93, 108;
  21, 38, 55,  72,  89, 106, 123, 140;
  24, 43, 62,  81, 100, 119, 138, 157, 176;
  27, 48, 69,  90, 111, 132, 153, 174, 195, 216;
  30, 53, 76,  99, 122, 145, 168, 191, 214, 237, 260;
  33, 58, 83, 108, 133, 158, 183, 208, 233, 258, 283, 308;
  36, 63, 90, 117, 144, 171, 198, 225, 252, 279, 306, 333, 360;
  etc.
the values of (2*h*k + k + h - 4) with h >= k >= 1. - _Vincenzo Librandi_, Jan 16 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A155724, A155722.

[n: n in [0..115]| not IsPrime(2*n+9)]; // Vincenzo Librandi, Mar 01 2012

Select[Range[0,80],!PrimeQ[2*#+9]&] (* Vincenzo Librandi, Mar 01 2012 *)

A162261 a(n) = (2n^3 + 5n^2 - 7*n)/2.

Original entry on oeis.org

0, 11, 39, 90, 170, 285, 441, 644, 900, 1215, 1595, 2046, 2574, 3185, 3885, 4680, 5576, 6579, 7695, 8930, 10290, 11781, 13409, 15180, 17100, 19175, 21411, 23814, 26390, 29145, 32085, 35216, 38544, 42075, 45815, 49770, 53946, 58349, 62985, 67860

Offset: 1

Vincenzo Librandi, Jun 29 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A151675, A155724.

[(2*n^3 + 5*n^2 - 7*n)/2 : n in [1..50]]; // Wesley Ivan Hurt, May 07 2021

CoefficientList[Series[x*(11-5*x)/(1-x)^4,{x,0,40}],x] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 11, 39, 90}, 50](* Vincenzo Librandi, Mar 04 2012 *)

def A162261(n): return n*(2*pow(n,2) +5*n -7)//2
print([A162261(n) for n in range(1,61)]) # G. C. Greubel, Jan 21 2025

Row sums from A155724: a(n) = Sum_{m=1..n} (2*m*n + m + n - 4).
From Vincenzo Librandi, Mar 04 2012: (Start)
G.f.: x^2*(11 - 5*x)/(1-x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
a(n) = A151675(n) - 8*n. - L. Edson Jeffery, Oct 12 2012
From Amiram Eldar, Feb 25 2023: (Start)
Sum_{n>=2} 1/a(n) = 8*log(2)/63 + 1166/19845.
Sum_{n>=2} (-1)^n/a(n) = (32*log(2) - 2*Pi - 3566/315)/63. (End)
E.g.f.: (1/2)*x^2*(11 + 2*x)*exp(x). - G. C. Greubel, Jan 21 2025

New name from Vincenzo Librandi, Mar 04 2012

A324937 Triangle read by rows: T(n, k) = 2nk + 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

Vincenzo Librandi, Mar 25 2019

			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.

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).

[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]];

t[n_, k_]:=2 n k + n + k - 8; Table[t[n, k], {n, 11}, {k, n}]//Flatten

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

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A155722 Numbers k such that 2*k + 9 is prime.

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A154685 Triangle read by rows: T(n, k) = 2*n*k + n + k + 4.

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A155723 Numbers k such that 2*k + 9 is not prime.

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A162261 a(n) = (2*n^3 + 5*n^2 - 7*n)/2.

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

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Formula

A154685 Triangle read by rows: T(n, k) = 2nk + n + k + 4.

A162261 a(n) = (2n^3 + 5n^2 - 7*n)/2.

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