A153039 -id:A153039 - cp's OEIS frontend

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

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

A153040 Numbers n>3 such that 2*n-5 is not a prime.

Original entry on oeis.org

7, 10, 13, 15, 16, 19, 20, 22, 25, 27, 28, 30, 31, 34, 35, 37, 40, 41, 43, 45, 46, 48, 49, 50, 52, 55, 58, 60, 61, 62, 63, 64, 65, 67, 69, 70, 73, 74, 75, 76, 79, 80, 82, 83, 85, 87, 88, 90, 91, 94, 95, 96, 97, 100, 103, 104, 105, 106, 107, 109, 110, 111

Offset: 1

Vincenzo Librandi, Dec 17 2008

One less than the associated number in A153039; one more than that in A153043. - R. J. Mathar Dec 20 2008

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

Cf. A089253, A153039.

[n: n in [4..120]|not IsPrime(2*n-5)]; // Vincenzo Librandi, Aug 30 2012

Select[Range[4,200],!PrimeQ[2*#-5]&] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2012 *)

Let p = prime number n = (p^2+5)/2 mod (p)

Flipped sign in definition. - R. J. Mathar, Dec 20 2008

A153041 Numbers n >=10 such that 2*n-19 is not a prime.

Original entry on oeis.org

10, 14, 17, 20, 22, 23, 26, 27, 29, 32, 34, 35, 37, 38, 41, 42, 44, 47, 48, 50, 52, 53, 55, 56, 57, 59, 62, 65, 67, 68, 69, 70, 71, 72, 74, 76, 77, 80, 81, 82, 83, 86, 87, 89, 90, 92, 94, 95, 97, 98, 101, 102, 103, 104

Offset: 1

Vincenzo Librandi, Dec 17 2008

One more than associated values in A153051, two more than A153047. - R. J. Mathar, Jan 05 2011

The terms after a(1) are the values of 2*h*k + k + h + 10, where h and k are positive integers.- Vincenzo Librandi, Jan 19 2013

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

Cf. A097932, A155704.
Numbers n such that 2n-k is not prime: A104275 (k=1), A153043 (k=3), A153040 (k=5), A153039 (k=7), A153044 (k=9), A153045 (k=11), A153049 (k=13), A153047 (k=15), A153051 (k=17), A153041 (k=19).
Similar sequence listed also in A089559, A153144.

[n: n in [10..150] | not IsPrime(2*n - 19)]; // Vincenzo Librandi, Jan 19 2013

Select[Range[10, 200], !PrimeQ[2 # - 19] &] (* Vincenzo Librandi, Jan 19 2013 *)

Edited by N. J. A. Sloane, Jun 22 2010

A153037 a(n) = 2n^2 + 16n + 23.

Original entry on oeis.org

23, 41, 63, 89, 119, 153, 191, 233, 279, 329, 383, 441, 503, 569, 639, 713, 791, 873, 959, 1049, 1143, 1241, 1343, 1449, 1559, 1673, 1791, 1913, 2039, 2169, 2303, 2441, 2583, 2729, 2879, 3033, 3191, 3353, 3519, 3689, 3863, 4041, 4223, 4409, 4599, 4793, 4991, 5193

Offset: 0

Vincenzo Librandi, Jan 25 2009

Sixth diagonal of triangle A154685.

Numbers of the form 2*k^2 - 9. - Bruno Berselli, Oct 30 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A153039, A154685.

I:=[23, 41, 63]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 22 2012

Table[2*n^2 + 16*n + 23, {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Feb 03 2012 *)
LinearRecurrence[{3, -3, 1}, {23, 41, 63}, 50] (* Vincenzo Librandi, Feb 22 2012 *)
CoefficientList[Series[(23 - 28*x  +9*x^2)/(1 -x)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Jan 04 2013 *)

a(n)=2*n^2+16*n+23 \\ Charles R Greathouse IV, Jan 11 2012

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 22 2012
G.f.: (23 - 28*x + 9*x^2)/(1-x)^3. - Vincenzo Librandi, Jan 04 2013
From Amiram Eldar, Feb 25 2023: (Start)
Sum_{n>=0} 1/a(n) = 137/126 - cot(3*Pi/sqrt(2))*Pi/(6*sqrt(2)).
Sum_{n>=0} (-1)^n/a(n) = 43/42 - cosec(3*Pi/sqrt(2))*Pi/(6*sqrt(2)). (End)
E.g.f.: exp(x)*(23 + 18*x + 2*x^2). - Elmo R. Oliveira, Feb 08 2025

Erroneously duplicated terms removed by Vincenzo Librandi, Feb 22 2012

A153044 Numbers n such that 2*n-9 is not a prime.

Original entry on oeis.org

5, 9, 12, 15, 17, 18, 21, 22, 24, 27, 29, 30, 32, 33, 36, 37, 39, 42, 43, 45, 47, 48, 50, 51, 52, 54, 57, 60, 62, 63, 64, 65, 66, 67, 69, 71, 72, 75, 76, 77, 78, 81, 82, 84, 85, 87, 89, 90, 92, 93, 96, 97, 98, 99, 102, 105, 106, 107, 108, 109, 111, 112, 113, 114, 115, 117, 120

Offset: 1

Vincenzo Librandi, Dec 17 2008

One more than the associated entry in A153039. - R. J. Mathar, Jan 05 2011

The terms after a(1) are the values of 2*h*k + k + h + 5, where h and k are positive integers. - Vincenzo Librandi, Jan 19 2013

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

Cf. A097069, A153039.

[n: n in [5..121] | not IsPrime(2*n-9)]; // Bruno Berselli, Mar 05 2011

Select[Range[150],!PrimeQ[2#-9]&]  (* Harvey P. Dale, Mar 05 2011 *)

Corrected and extended by N. J. A. Sloane, May 29 2010

cp's OEIS Frontend

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

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A153040 Numbers n>3 such that 2*n-5 is not a prime.

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A153041 Numbers n >=10 such that 2*n-19 is not a prime.

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Magma

Mathematica

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A153037 a(n) = 2*n^2 + 16*n + 23.

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Programs

Magma

Mathematica

PARI

Formula

Extensions

A153044 Numbers n such that 2*n-9 is not a prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Extensions

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

A153037 a(n) = 2n^2 + 16n + 23.