A153238 -id:A153238 - cp's OEIS frontend

A144562 Triangle read by rows: T(n, k) = 2nk + n + k - 1.

3, 6, 11, 9, 16, 23, 12, 21, 30, 39, 15, 26, 37, 48, 59, 18, 31, 44, 57, 70, 83, 21, 36, 51, 66, 81, 96, 111, 24, 41, 58, 75, 92, 109, 126, 143, 27, 46, 65, 84, 103, 122, 141, 160, 179, 30, 51, 72, 93, 114, 135, 156, 177, 198, 219, 33, 56, 79, 102, 125, 148, 171, 194, 217, 240, 263

Offset: 1

Vincenzo Librandi, Jan 06 2009

Rearrangement of A153238, numbers n such that 2*n+3 is not prime (we have 2*T(n,k) + 3 = (2*n+1)*(2*k+1), as 2*n+3 is odd it consists of (at least) two odd factors and all such factors appear by definition).

			Triangle begins:
   3;
   6, 11;
   9, 16, 23;
  12, 21, 30, 39;
  15, 26, 37, 48,  59;
  18, 31, 44, 57,  70,  83;
  21, 36, 51, 66,  81,  96, 111;
  24, 41, 58, 75,  92, 109, 126, 143;
  27, 46, 65, 84, 103, 122, 141, 160, 179;
  ...

Vincenzo Librandi, Rows n = 1..100, flattened
Mutsumi Suzuki Vincenzo Librandi's method for sequential primes (Librandi's description in Italian).

Cf. A067076, A144640, A153238.
Main diagonal gives A142463.
T(2n,n) gives A180863(n+1).

[2*n*k+n+k-1: k in [1..n], n in [1..11]]; /* or, see example: */ [[2*n*k+n+k-1: k in [1..n]]: n in [1..9]]; // Bruno Berselli, Dec 04 2011

A144562:= (n,k) -> 2*n*k +n +k -1; seq(seq(A144562(n,k), k=1..n), n=1..12); # G. C. Greubel, Mar 01 2021

T[n_, k_]:= 2*n*k +n +k -1; Table[T[n, k], {n, 11}, {k, n}]//Flatten

T(n,k)=2*n*k+n+k-1 \\ Charles R Greathouse IV, Dec 28 2011

flatten([[2*n*k+n+n-1 for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 01 2021

Sum_{k=1..n} T(n,k) = n*(2*n^2 + 5*n - 1)/2 = A144640(n). - G. C. Greubel, Mar 01 2021

G.f.: x*y*(3 + 2*x*y + 2*x^3*y^2 - x^2*y*(6 + y))/((1 - x)^2*(1 - x*y)^3). - Stefano Spezia, Nov 04 2024

Edited by Ray Chandler, Jan 07 2009

A139606 a(n) = 15*n + 6.

Original entry on oeis.org

6, 21, 36, 51, 66, 81, 96, 111, 126, 141, 156, 171, 186, 201, 216, 231, 246, 261, 276, 291, 306, 321, 336, 351, 366, 381, 396, 411, 426, 441, 456, 471, 486, 501, 516, 531, 546, 561, 576, 591, 606, 621, 636, 651, 666, 681, 696, 711, 726, 741, 756, 771, 786

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 6th column of positive numbers in the square array of nonnegative and polygonal numbers A139600.

6th transversal numbers (or 6-transversal numbers): (A000217(6)-6)*n + 6.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 189. - From N. J. A. Sloane, Dec 01 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A067076, A144562, A153238.

Cf. A000217, A008597, A016873, A057145, A139600.

[15*n+6 : n in [0..60]]; // Vincenzo Librandi, Oct 06 2011

Range[6, 1000, 15] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
15*Range[0,60]+6 (* or *) LinearRecurrence[{2,-1},{6,21},60] (* Harvey P. Dale, Aug 09 2019 *)

a(n)=15*n+6 \\ Charles R Greathouse IV, Oct 05 2011

a(n) = A057145(n+2,6).
G.f.: 3*(2+3*x)/(x-1)^2 . - R. J. Mathar, Jul 28 2016
From Elmo R. Oliveira, Apr 12 2024: (Start)
E.g.f.: 3*exp(x)*(2 + 5*x).
a(n) = 3*A016873(n) = A008597(n) + 6.
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

A098828 Primes of the form 2n^2 + 2n - 1.

Original entry on oeis.org

3, 11, 23, 59, 83, 179, 263, 311, 419, 479, 683, 839, 1103, 1511, 2111, 2243, 2663, 2963, 3119, 4139, 4703, 5099, 5303, 5939, 7079, 10223, 11399, 12011, 12323, 12959, 17483, 19403, 21011, 21839, 22259, 24419, 25763, 27143, 27611, 28559, 30011

Offset: 1

Giovanni Teofilatto, Oct 09 2004

nonn

a(n)==3 (mod 4).
Equivalently primes p such that 2p+3 is square.
Also 3 followed by primes p of the form 2*n^2+6*n+3 = 2*(n+2)^2-2*(n+2)-1 (see the first comment) such that 2^(p-1)+3 is not prime. - Vincenzo Librandi, Jan 03 2009; M. F. Hasler, Jan 07 2009; R. J. Mathar, Jan 14 2009; Bruno Berselli, Sep 23 2013

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

Cf. A109358, A109367, A153238

[3] cat [ p: p in PrimesUpTo(30100) | exists(t){ n: n in [1..Isqrt(p div 2)] | 2*n^2+6*n+3 eq p } and not IsPrime(2^(p-1)+3) ];

Select[Table[Prime[n], {n, 3500}], IntegerQ[(2# + 3)^(1/2)] &] (* Ray Chandler, Oct 26 2004 *)

list(lim)=my(v=List()); for(n=1,oo, my(t=2*n*(n+1)-1); if(t>lim, return(Vec(v))); if(isprime(t), listput(v,t))) \\ Charles R Greathouse IV, Feb 26 2025

a(n) = (A109367(n) - 3)/2.

Corrected by Ray Chandler, Oct 26 2004
Edited by N. J. A. Sloane, Jan 25 2009
Name edited by Charles R Greathouse IV, Feb 26 2025

A111199 Numbers k such that 4k + 9 is prime.

Original entry on oeis.org

1, 2, 5, 7, 8, 11, 13, 16, 20, 22, 23, 25, 26, 32, 35, 37, 41, 43, 46, 47, 55, 56, 58, 62, 65, 67, 68, 71, 76, 77, 82, 85, 86, 91, 95, 97, 98, 100, 103, 106, 110, 112, 113, 125, 128, 133, 137, 140, 142, 146, 148, 151, 152, 158, 161, 163, 166, 167, 173, 175, 181, 187

Offset: 1

Parthasarathy Nambi, Oct 24 2005

nonn

			For k=98, 4*k + 9 = 401 (prime).

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A005098, A095278, A089986, A153238, A144562.

is(n)=isprime(4*n+9) \\ Charles R Greathouse IV, Feb 20 2017

a(n) = A005098(n+1) - 2. - R. J. Mathar, Sep 23 2009

More terms from R. J. Mathar, Sep 23 2009

A153144 Numbers n such that 2*n+19 is not a prime.

Original entry on oeis.org

1, 3, 4, 7, 8, 10, 13, 15, 16, 18, 19, 22, 23, 25, 28, 29, 31, 33, 34, 36, 37, 38, 40, 43, 46, 48, 49, 50, 51, 52, 53, 55, 57, 58, 61, 62, 63, 64, 67, 68, 70, 71, 73, 75, 76, 78, 79, 82, 83, 84, 85, 88, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101

Offset: 1

Vincenzo Librandi, Dec 19 2008

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

Cf. A153143, A153081, A155551.

Numbers n such that 2n+k is not prime: A047845 (k=1), A153238 (k=3), A153052 (k=5), A153053 (k=7), A153723 (k=9), A153083 (k=11), A153082 (k=13), A241571 (k=15), A241572 (k=17), this sequence (k=19).

[n: n in [1..120] | not IsPrime(2*n + 19)]; // Vincenzo Librandi, Dec 13 2012

Select[Range[0, 500], !PrimeQ[2# + 19] &] (* Vincenzo Librandi, Dec 13 2012 *)

A153052 Numbers m such that 2*m + 5 is not a prime.

Original entry on oeis.org

2, 5, 8, 10, 11, 14, 15, 17, 20, 22, 23, 25, 26, 29, 30, 32, 35, 36, 38, 40, 41, 43, 44, 45, 47, 50, 53, 55, 56, 57, 58, 59, 60, 62, 64, 65, 68, 69, 70, 71, 74, 75, 77, 78, 80, 82, 83, 85, 86, 89, 90, 91, 92, 95, 98, 99, 100, 101, 102, 104, 105, 106, 107, 108, 110

Offset: 1

Vincenzo Librandi, Dec 17 2008

One less than the associated entry in A153238. - R. J. Mathar, Jan 05 2011

The terms are the values of 2*h*k + k + h - 2, where h and k are positive integers. - Vincenzo Librandi, Jan 19 2013

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

Cf. A089038, A153238, A154680.

[n: n in [0..110]| not IsPrime(2*n+5)]; // Vincenzo Librandi, Oct 16 2012

Select[Range[0, 150], !PrimeQ[2*# + 5] &] (* Vincenzo Librandi, Oct 16 2012 *)

A163769 Primes p such that 2*p+3 is not prime.

Original entry on oeis.org

3, 11, 23, 31, 37, 41, 59, 61, 71, 79, 83, 101, 103, 107, 109, 131, 149, 151, 163, 179, 181, 191, 211, 233, 239, 241, 251, 257, 263, 271, 281, 293, 311, 313, 317, 331, 347, 359, 367, 373, 389, 401, 419, 421, 431, 433, 443, 449, 457, 461, 479, 491, 499, 521

Offset: 1

Vincenzo Librandi, Aug 04 2009

All those p appear in A144562. [Proof: since 2p+3 is odd and not prime, it can be written as a product of two odd numbers, 2p+3=(2k+1)*(2s+1), therefore p=2ks+k+s-1. - R. J. Mathar, Aug 06 2009]

			3 is in the sequence because 2*3+3=9 is composite; 23 is in the sequence because 2*23+3=49 is composite.

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

Cf. A144562.

[p: p in PrimesUpTo(700) | not IsPrime(2*p+3)]; // Vincenzo Librandi, Apr 08 2013

Select[Prime[Range[200]],!PrimeQ[2#+3]&] (* Harvey P. Dale, Feb 02 2012 *)

A153238 INTERSECT A000040. - R. J. Mathar, Aug 05 2009

A000040 \ A023204. - R. J. Mathar, Aug 05 2009

Entries checked - R. J. Mathar, Aug 06 2009

A165635 Primes of the form (p^2 - 3)/2 where p is also prime.

Original entry on oeis.org

3, 11, 23, 59, 83, 179, 263, 419, 479, 683, 839, 1103, 2243, 2663, 3119, 4703, 5099, 5303, 5939, 11399, 12323, 19403, 22259, 25763, 27143, 28559, 33023, 34583, 42923, 47123, 54779, 56783, 60899, 62303, 64439, 67343, 75659, 78803, 83639, 98123

Offset: 1

Vincenzo Librandi, Sep 23 2009

The sequence could be generated by searching for squared primes p^2 in A153238.

			The prime 3=(3^2-3)/2 is generated by p=3. The prime 11=(5^2-3)/2 is generated by p=5. The prime 23 by p=7.

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

Cf. A110589, A153238.

[a: p in PrimesInInterval(1, 500) | IsPrime(a) where a is (p^2 - 3) div 2]; // Vincenzo Librandi, Oct 12 2012

Select[Table[(p^2 - 3)/2, {p, Prime[Range[300]]}], PrimeQ] (* Vincenzo Librandi, Oct 12 2012 *)

a(n) = (A110589(n)^2-3)/2 .

More terms from Max Alekseyev, Sep 25 2009

Comment clarified by R. J. Mathar, Oct 07 2009

A282026 a(n) is the smallest m with gcd(m, 2n+1) = 1 such that 2n + 2*m + 1 is composite.

Original entry on oeis.org

4, 11, 2, 1, 8, 2, 1, 17, 2, 1, 2, 1, 1, 4, 2, 1, 1, 2, 1, 5, 2, 1, 2, 1, 1, 2, 1, 1, 4, 2, 1, 1, 2, 1, 4, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 8, 2, 1, 8, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 4, 2, 1, 1, 1, 1, 4, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1

Offset: 0

N. J. A. Sloane, Feb 12 2017

nonn

Starting at 2*n + 1, find the next odd composite number 2*n + 2*m + 1 that is relatively prime to 2*n + 1; then a(n) = m.
Since 2*n + 3 is relatively prime to 2*n + 1, and (2*n + 3)^2 is composite, a(n) <= 2*n^2 + 5*n + 4 (this is tight for n=0 and n=1).
From Andrey Zabolotskiy, Feb 13 2017: (Start)
Up to n = 10^7, a(n) are from the set [1, 2, 4, 5, 7, 8, 11, 13, 14, 16, 17, 19, 22]. First occurrence of 14 is a(99412), first occurrence of 22 is a(7225627). [Thanks to Altug Alkan for pointing out a(99412).]
a(n) = 1 iff n is in A153238.
(End)
Based on Altug Alkan's b-file, the records in this sequence are 4, 11, 17, 19, ... and occur at positions 1, 2, 8, 638, ... If the sequence is unbounded, then these two subsidiary sequences should be added to the OEIS (if they are new). - N. J. A. Sloane, Feb 13 2017

			When n=1, 2*n + 1 = 3, and 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 are all either prime or have a common factor with 3. The next term, 25, is OK, and so a(1) = (25 - 3)/2 = 11.

Altug Alkan, Table of n, a(n) for n = 0..10000

Cf. A153238, A282423, A282429.

Table[m = 1; While[Nand[CoprimeQ[m, 2 n + 1], CompositeQ[2 (n + m) + 1]], m++]; m, {n, 0, 120}] (* Michael De Vlieger, Feb 18 2017 *)

a(n) = my(k=1); while(isprime(2*n+2*k+1) || gcd(2*n+1, k) != 1, k++); k; \\ Altug Alkan, Feb 13 2017

Definition corrected by Altug Alkan, Feb 13 2017

A154591 a(n) = 2n^2 + 18n + 7.

Original entry on oeis.org

27, 51, 79, 111, 147, 187, 231, 279, 331, 387, 447, 511, 579, 651, 727, 807, 891, 979, 1071, 1167, 1267, 1371, 1479, 1591, 1707, 1827, 1951, 2079, 2211, 2347, 2487, 2631, 2779, 2931, 3087, 3247, 3411, 3579, 3751, 3927, 4107, 4291, 4479, 4671, 4867, 5067, 5271

Offset: 1

Vincenzo Librandi, Jan 12 2009

Ninth diagonal of A144562.

2*a(n) + 67 is a square.

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

Cf. A067076, A144562, A153238.

I:=[27, 51, 79]; [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

LinearRecurrence[{3, -3, 1}, {27, 51, 79}, 50] (* Vincenzo Librandi, Feb 22 2012 *)

for(n=1, 40, print1(2*n^2 + 18*n + 7", ")); \\ Vincenzo Librandi, Feb 22 2012

[2*n^2+18*n+7 for n in range(1,51)] #  G. C. Greubel, May 27 2024

G.f.: (9*x^2-6*x-7)/(x-1)^3. - Bruno Berselli, Dec 07 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 22 2012
Sum_{n>=1} 1/a(n) = 1621/20097 + tan(sqrt(67)*Pi/2)*Pi/(2*sqrt(67)). - Amiram Eldar, Feb 25 2023
E.g.f.: (7 + 20*x + 2*x^2)*exp(x). - G. C. Greubel, May 27 2024

cp's OEIS Frontend

A144562 Triangle read by rows: T(n, k) = 2*n*k + n + k - 1.

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A139606 a(n) = 15*n + 6.

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A098828 Primes of the form 2*n^2 + 2*n - 1.

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Magma

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A111199 Numbers k such that 4k + 9 is prime.

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PARI

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A153144 Numbers n such that 2*n+19 is not a prime.

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Magma

Mathematica

A153052 Numbers m such that 2*m + 5 is not a prime.

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Magma

Mathematica

A163769 Primes p such that 2*p+3 is not prime.

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A144562 Triangle read by rows: T(n, k) = 2nk + n + k - 1.

A098828 Primes of the form 2n^2 + 2n - 1.

A282026 a(n) is the smallest m with gcd(m, 2n+1) = 1 such that 2n + 2*m + 1 is composite.

A154591 a(n) = 2n^2 + 18n + 7.