A089038 -id:A089038 - cp's OEIS frontend

A023243 Primes that remain prime through 2 iterations of the function f(x) = 2x + 5.

7, 13, 31, 37, 67, 73, 79, 139, 151, 181, 367, 541, 613, 661, 709, 739, 787, 811, 829, 997, 1087, 1117, 1123, 1249, 1327, 1669, 1753, 1801, 1861, 1999, 2011, 2113, 2179, 2239, 2293, 2473, 2557, 2659, 2713, 2719, 3037, 3181, 3187, 3271, 3301, 3517, 3727, 3793

Offset: 1

David W. Wilson

Primes p such that 2*p+5 and 4*p+15 are also primes. - Vincenzo Librandi, Aug 04 2010

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of A023205 and A089038.

[n: n in [0..100000] | IsPrime(n) and IsPrime(2*n+5) and IsPrime(4*n+15)] // Vincenzo Librandi, Aug 04 2010

is(n)=n%6==1 && isprime(2*n+5) && isprime(4*n+15) && isprime(n) \\ Charles R Greathouse IV, Sep 12 2016

a(n) == 1 (mod 6). - John Cerkan, Sep 12 2016

A182138 Irregular triangle T, read by rows, in which row n lists the distances between n and the two primes whose sum makes 2n in decreasing order (Goldbach conjecture).

Original entry on oeis.org

0, 0, 1, 2, 0, 1, 4, 0, 5, 3, 4, 2, 7, 3, 8, 6, 0, 7, 5, 1, 10, 6, 0, 9, 3, 8, 4, 2, 13, 3, 14, 12, 6, 0, 13, 11, 5, 1, 12, 0, 17, 9, 3, 16, 10, 8, 2, 19, 15, 9, 20, 18, 6, 0, 19, 17, 13, 7, 5, 22, 18, 12, 6, 21, 15, 3, 20, 16, 14, 10, 4, 25, 15, 9, 24, 18, 12, 0, 23, 17, 13, 11, 7, 1

Offset: 2

Jean COHEN, Apr 16 2012

The Goldbach conjecture is that for any even integer 2n>=4, at least one pair of primes p and q exist such that p+q=2n. The present numbers listed here are the distances d between each prime and n, the half of the even integer 2n: d=n-p=q-n with p <= q.
See the link section for plots I added. - Jason Kimberley, Oct 04 2012
Each nonzero entry d of row n is coprime to n. For otherwise n+d would be composite. - Jason Kimberley, Oct 10 2012

			n=2, 2n=4, 4=2+2, p=q=2 -> d=0.
n=18, 2n=36, four prime pairs have a sum of 36: 5+31, 7+29, 13+23, 17+19, with the four distances d being 13=18-5=31-18, 11=18-7=29-18, 5=18-13=23-18, 1=18-17=19-18.
Triangle begins:
  0;
  0;
  1;
  2, 0;
  1;
  4, 0;
  5, 3;
  4, 2;
  7, 3;
  8, 6, 0;

Alois P. Heinz, Rows n = 2..600, flattened
OEIS (Plot 2), Plot of (n, d)
Subplots for fixed p:
OEIS (Plot 2), A067076 vs A098090 (p=3).
OEIS (Plot 2), A089038 vs A089253 (p=5).
OEIS (Plot 2), A105760 vs A089192 (p=7).
...
OEIS (Plot 2), A153143 vs A097932 (p=19).
Wikipedia, Goldbach's conjecture

Cf. A045917 (row lengths), A047949 (first column), A047160 (last elements of rows).

Cf. A184995.

A182138:= func;
&cat[A182138(n):n in [2..30]]; // Jason Kimberley, Oct 01 2012

T:= n-> seq(`if`(isprime(p) and isprime(2*n-p), n-p, NULL), p=2..n):
seq(T(n), n=2..40); # Alois P. Heinz, Apr 16 2012

T[n_] := Table[If[PrimeQ[p] && PrimeQ[2n-p], n-p, Nothing], {p, 2, n}];
Table[T[n], {n, 2, 30}] // Flatten (* Jean-François Alcover, Jan 09 2025, after Alois P. Heinz *)

for(n=2,18,forprime(p=2,n,if(isprime(2*n-p),print1(n-p", ")))) \\ Charles R Greathouse IV, Apr 16 2012

T(n,i) = n - A184995(n,i). - Jason Kimberley, Sep 25 2012

A023274 Primes that remain prime through 3 iterations of function f(x) = 2x + 5.

Original entry on oeis.org

13, 31, 37, 67, 73, 181, 367, 541, 661, 997, 1087, 1117, 1327, 1861, 2179, 2293, 2473, 2713, 2719, 3271, 3727, 4363, 5281, 5443, 5749, 7459, 8089, 8707, 9109, 9181, 9337, 10357, 10639, 12553, 13183, 14923, 16183, 16249, 17341, 17419, 17761, 17923, 17989

Offset: 1

David W. Wilson

nonn

Primes p such that 2*p+5, 4*p+15 and 8*p+35 are also primes. - Vincenzo Librandi, Aug 04 2010

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of A023205, A023243, and of A089038.

[n: n in [1..100000] | IsPrime(n) and IsPrime(2*n+5) and IsPrime(4*n+15) and IsPrime(8*n+35)] // Vincenzo Librandi, Aug 04 2010

select(t -> isprime(t) and isprime(2*t+5) and isprime(4*t+15) and isprime(8*t+35), [seq(t,t=7..20000,6)]);# Robert Israel, Sep 18 2016

Select[Prime@ Range@ 2100, Times @@ Boole@ PrimeQ@ Rest@ NestList[2 # + 5 &, #, 3] > 0 &] (* Michael De Vlieger, Sep 16 2016 *)

a(n) == 1 (mod 6). - John Cerkan, Sep 16 2016

A073273 a(n) = floor(sqrt(prime(n)*prime(n+2))).

Original entry on oeis.org

3, 4, 7, 9, 13, 15, 19, 23, 26, 32, 35, 39, 43, 47, 52, 56, 62, 65, 69, 74, 77, 83, 89, 94, 99, 103, 105, 109, 117, 121, 131, 134, 142, 144, 152, 156, 161, 167, 172, 176, 184, 186, 193, 195, 203, 210, 218, 225, 229, 233, 236, 244, 248, 256, 262, 266, 272, 275

Offset: 1

Reinhard Zumkeller, Jul 22 2002

nonn

A000040(n) < a(n) < A000040(n+2).

			prime(10)*prime(12) = 29*37 = 1073 = 32*32+49, therefore a(10)=32; A073274(10) = prime(11)-a(10) = 31-32 = -1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000040, A073271, A073274, A078444.

[Floor(Sqrt(NthPrime(n)*NthPrime(n+2))): n in [1..60]]; // Vincenzo Librandi, Dec 12 2015

Table[Floor[Sqrt[Prime[n] Prime[n + 2]]], {n, 60}] (* Vincenzo Librandi, Dec 12 2015 *)

a(n) = sqrtint(prime(n)*prime(n+2)); \\ Michel Marcus, Dec 12 2015

a(n) = A098090(A028310(n - 1)) + A089038(n). - Miko Labalan, Dec 12 2015

A023304 Primes that remain prime through 4 iterations of function f(x) = 2x + 5.

Original entry on oeis.org

13, 31, 181, 541, 661, 1087, 1861, 2179, 2719, 3727, 7459, 8089, 8707, 9109, 10639, 17341, 19333, 22501, 23293, 29287, 32797, 39847, 40387, 42703, 46591, 51613, 53101, 56149, 56809, 57829, 59233, 80779, 87643, 89113, 89413, 91291, 93979, 94261, 98899

Offset: 1

David W. Wilson

nonn

Primes p such that 2*p+5, 4*p+15, 8*p+35 and 16*p+75 are also primes. - Vincenzo Librandi, Aug 04 2010

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of A023205, A023243, A023274, and A089038.

[n: n in [1..1000000] | IsPrime(n) and IsPrime(2*n+5) and IsPrime(4*n+15) and IsPrime(8*n+35) and IsPrime(16*n+75)]; // Vincenzo Librandi, Aug 04 2010

Select[Prime@ Range[10^4], Times @@ Boole@ PrimeQ@ Rest@ NestList[2 # + 5 &, #, 4] > 0 &] (* Michael De Vlieger, Sep 27 2016 *)
Select[Prime[Range[10000]],AllTrue[Rest[NestList[2#+5&,#,4]],PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 23 2020 *)

a(n) == 1 (mod 6). - John Cerkan, Sep 27 2016

Definition clarified by Harvey P. Dale, Oct 23 2020

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 *)

A154680 Triangle read by rows where T(m,n)=2mn + m + n - 2.

Original entry on oeis.org

2, 5, 10, 8, 15, 22, 11, 20, 29, 38, 14, 25, 36, 47, 58, 17, 30, 43, 56, 69, 82, 20, 35, 50, 65, 80, 95, 110, 23, 40, 57, 74, 91, 108, 125, 142, 26, 45, 64, 83, 102, 121, 140, 159, 178, 29, 50, 71, 92, 113, 134, 155, 176, 197, 218, 32, 55, 78, 101, 124, 147, 170, 193, 216, 239, 262

Offset: 1

Vincenzo Librandi, Jan 18 2009

All terms are in A153052.

First column: A016789; second column: 5*A000027; third column: A016993; fourth column: A017185. - Vincenzo Librandi, Nov 18 2012

			Triangle begins:
2;
5,  10;
8,  15, 22;
11, 20, 29, 38;
14, 25, 36, 47, 58;
17, 30, 43, 56, 69,  82;
20, 35, 50, 65, 80,  95,  110;
23, 40, 57, 74, 91,  108, 125, 142;
26, 45, 64, 83, 102, 121, 140, 159, 178;
29, 50, 71, 92, 113, 134, 155, 176, 197, 218; etc.

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

Cf. A153052, A089038, A016789, A000027, A017185.

[2*n*k+n+k-2: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 18 2012

Flatten[Table[Floor[2 n m + m + n - 2], {n, 1, 16}, {m, n}]] (* Vincenzo Librandi, May 14 2012 *)

A023332 Primes that remain prime through 5 iterations of function f(x) = 2x + 5.

Original entry on oeis.org

13, 541, 1087, 1861, 3727, 23293, 40387, 87643, 98899, 109111, 115153, 116329, 119101, 131617, 133597, 163909, 197521, 214021, 215389, 218227, 238207, 263239, 294751, 489901, 493693, 665527, 734131, 767881, 808693, 895351, 1038127, 1051957

Offset: 1

David W. Wilson

nonn

Primes p such that 2*p+5, 4*p+15, 8*p+35, 16*p+75 and 32*p+155 are also primes. - Vincenzo Librandi, Aug 04 2010

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of A023205, A023243, A023274, A023304, and A089038.

[n: n in [1..5000000] | IsPrime(n) and IsPrime(2*n+5) and IsPrime(4*n+15) and IsPrime(8*n+35) and IsPrime(16*n+75) and IsPrime(32*n+155)] // Vincenzo Librandi, Aug 04 2010

txQ[p_]:=AllTrue[NestList[2#+5&,p,5],PrimeQ]; Select[Prime[Range[83000]],txQ] (* Harvey P. Dale, May 10 2024 *)

a(n) == 1 (mod 6). - John Cerkan, Oct 09 2016

A086304 Numbers n such that n+6 is prime.

Original entry on oeis.org

1, 5, 7, 11, 13, 17, 23, 25, 31, 35, 37, 41, 47, 53, 55, 61, 65, 67, 73, 77, 83, 91, 95, 97, 101, 103, 107, 121, 125, 131, 133, 143, 145, 151, 157, 161, 167, 173, 175, 185, 187, 191, 193, 205, 217, 221, 223, 227, 233, 235, 245, 251, 257, 263, 265, 271, 275, 277

Offset: 1

Giovanni Teofilatto, Aug 29 2003

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano 1997

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

Cf. A089038.

[n: n in [0..300]| IsPrime(n + 6)]; // Vincenzo Librandi, Oct 16 2012

Select[Range[0, 900], PrimeQ[# + 6] &] (* Vincenzo Librandi, Oct 16 2012 *)

is(n)=isprime(n+6) \\ Charles R Greathouse IV, May 22 2017

a(n) = 2*A089038(n+1)-1.

Corrected and extended by Ray Chandler, Nov 29 2003

A337767 Array T(n,k) (n >= 1, k >= 1) read by upward antidiagonals and defined as follows. Let N(p,i) denote the result of applying "nextprime" i times to p; T(n,k) = smallest prime p such that N(p,n) - p = 2*k, or 0 if no such prime exists.

Original entry on oeis.org

3, 0, 7, 0, 3, 23, 0, 0, 5, 89, 0, 0, 0, 23, 139, 0, 0, 0, 3, 19, 199, 0, 0, 0, 0, 7, 47, 113, 0, 0, 0, 0, 3, 17, 83, 1831, 0, 0, 0, 0, 0, 5, 23, 211, 523, 0, 0, 0, 0, 0, 0, 17, 43, 109, 887, 0, 0, 0, 0, 0, 0, 3, 13, 79, 317, 1129, 0, 0, 0, 0, 0, 0, 0, 7, 19, 107, 619, 1669

Offset: 1

Robert G. Wilson v, Sep 19 2020

The positive entries in each row and column are distinct.
Number of zeros right of 3 are 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 3, 3, 3, 6, 5, 5, 4, 6, ..., .
Number of zeros in the n-th row are 0, 1, 3, 4, 6, 7, 10, 13, 14, 17, 18, 20, 22, 25, 28, 30, 32, 36, 37, 40, 45, 47, 51, 52, 55, ..., .
The usual convention in the OEIS is to use -1 in the "escape clause" - that is, when "no such terms exists". It is probably too late to change this sequence, but it should not be cited as a role model for other sequences. - N. J. A. Sloane, Jan 19 2021
a(1416), a(1637), and a(1753) were provided by Brian Kehrig. - Martin Raab, Jun 28 2024

			The initial rows of the array are:
  3, 7, 23, 89, 139, 199, 113, 1831, 523, 887, 1129, 1669, 2477, 2971, 4297, ...
  0, 3, 5, 23, 19, 47, 83, 211, 109, 317, 619,  199, 1373, 1123, 1627, 4751, ...
  0, 0, 0,  3,  7, 17, 23,  43,  79, 107, 109,  113,  197,  199,  317,  509, ...
  0, 0, 0,  0,  3,  5, 17,  13,  19,  47,  79,   73,  113,  109,  193,  317, ...
  0, 0, 0,  0,  0,  0,  3,   7,  11,  17,  19,   43,   71,   73,  107,  191, ...
  0, 0, 0,  0,  0,  0,  0,   3,   5,  11,   7,   13,   41,   31,   67,  107, ...
  0, 0, 0,  0,  0,  0,  0,   0,   0,   3,   0,    5,   11,   13,   23,   47, ...
  0, 0, 0,  0,  0,  0,  0,   0,   0,   0,   0,    0,    3,    0,    7,   29, ...
  0, 0, 0,  0,  0,  0,  0,   0,   0,   0,   0,    0,    0,    3,    0,    5, ...
The initial antidiagonals are:
  [3]
  [0, 7]
  [0, 3, 23]
  [0, 0, 5, 89]
  [0, 0, 0, 23, 139]
  [0, 0, 0, 3, 19, 199]
  [0, 0, 0, 0, 7, 47, 113]
  [0, 0, 0, 0, 3, 17, 83, 1831]
  [0, 0, 0, 0, 0, 5, 23, 211, 523]
  [0, 0, 0, 0, 0, 0, 17, 43, 109, 887]
  [0, 0, 0, 0, 0, 0, 3, 13, 79, 317, 1129]
  ...

Martin Raab, Table of n, a(n) for n = 1..1830 (antidiagonals 1..60)

Cf. A000230, A144103, A339943, A339944 (rows 1 to 4), A086153.

t[r_, c_] := If[ 2c <= Prime[r + 2] - 5, 0, Block[{p = 3}, While[ NextPrime[p, r] != 2c + p && p < 52000000, p = NextPrime@ p]; If[p > 52000000, 0, p]]]; Table[ t[r -c +1, c], {r, 11}, {c, r}] // Flatten

T(n,k) = 0 if prime(n+2)-5 <= 2k. A089038.

T(n,k) = 3 if prime(n+2) = 2k+6. A067076.

Entry revised by N. J. A. Sloane, Nov 07 2020

Deleted a-file and b-file because entries were unreliable. - N. J. A. Sloane, Nov 01 2021

cp's OEIS Frontend

A023243 Primes that remain prime through 2 iterations of the function f(x) = 2x + 5.

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A182138 Irregular triangle T, read by rows, in which row n lists the distances between n and the two primes whose sum makes 2n in decreasing order (Goldbach conjecture).

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A023274 Primes that remain prime through 3 iterations of function f(x) = 2x + 5.

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A073273 a(n) = floor(sqrt(prime(n)*prime(n+2))).

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A023304 Primes that remain prime through 4 iterations of function f(x) = 2x + 5.

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

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A154680 Triangle read by rows where T(m,n)=2*m*n + m + n - 2.

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A154680 Triangle read by rows where T(m,n)=2mn + m + n - 2.