A105760 -id:A105760 - cp's OEIS frontend

A153053 Numbers j such that 2*j + 7 is not a prime.

1, 4, 7, 9, 10, 13, 14, 16, 19, 21, 22, 24, 25, 28, 29, 31, 34, 35, 37, 39, 40, 42, 43, 44, 46, 49, 52, 54, 55, 56, 57, 58, 59, 61, 63, 64, 67, 68, 69, 70, 73, 74, 76, 77, 79, 81, 82, 84, 85, 88, 89, 90, 91, 94, 97, 98, 99, 100, 101, 103, 104, 105, 106, 107, 109, 112, 114, 115

Offset: 1

Vincenzo Librandi, Dec 17 2008

Let p = prime number, n = (p^2-7)/2 (mod p).

Comment: All numbers of the form 1+3k (k=0,1,2,...) are in this sequence, since 2(3k+1)+7 = 6k+9 is divisible by 3. Moreover, each of these numbers can be extended to an equidistant sequence of length k+1 and step 2k+3: This leads to the triangle T[k,m] = (3k+1)+(2k+3)*m, m=0,...,k, of elements of this sequence, because T[k,m]*2+7 = (2k+3)(2m+3) is never prime. The lines of the triangle end with m=k since the next term T[k,k+1] would be the same as the term in the following line, T[k+1,k]. (The formula T[k,m]=((2k+3)(2m+3)-7)/2 might also explain the comment involving "n=(p^2-7)/2".) [M. F. Hasler, Jun 16 2010]

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

Cf. A105760, A154684.

[n: n in [1..120] | not IsPrime(2*n + 7)]; // Vincenzo Librandi, Nov 21 2012

Select[Range[200], !PrimeQ[2# + 7] &] (* Vincenzo Librandi, Nov 21 2012 *)

for(n=1,200,isprime(2*n+7)||print1(n", ")) \\ M. F. Hasler, Jun 16 2010

Checked and extended by M. F. Hasler, Jun 16 2010

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

A023244 Primes that remain prime through 2 iterations of the function f(x) = 2x + 7.

Original entry on oeis.org

2, 5, 17, 23, 53, 83, 137, 197, 227, 257, 293, 317, 347, 383, 467, 593, 647, 677, 683, 797, 857, 953, 1163, 1193, 1217, 1607, 1877, 1907, 1913, 1997, 2063, 2207, 2237, 2843, 2903, 3023, 3257, 3323, 3557, 3947, 4133, 4253, 4517, 4583, 4643, 4967, 5087, 5387

Offset: 1

David W. Wilson

nonn

Primes p such that 2*p+7 and 4*p+21 are also primes. - Vincenzo Librandi, Aug 04 2010

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

Subsequence of A023206 and A105760.

[n: n in [0..100000] | IsPrime(n) and IsPrime(2*n+7) and IsPrime(4*n+21)] // Vincenzo Librandi, Aug 04 2010

Select[Prime@ Range[10^3], Times @@ Boole@ PrimeQ@ NestList[2 # + 7 &, #, 2] > 0 &] (* Michael De Vlieger, Sep 12 2016 *)

a(n) == 5 (mod 6), for n > 1. - John Cerkan, Sep 12 2016

A088767 a(n) = A087697(n)/2.

Original entry on oeis.org

5, 6, 12, 15, 18, 27, 30, 33, 45, 48, 60, 72, 78, 87, 93, 102, 117, 132, 135, 138, 150, 162, 180, 183, 195, 213, 225, 228, 258, 282, 285, 297, 300, 303, 312, 327, 333, 342, 363, 375, 390, 402, 408, 423, 435, 480, 492, 495, 513, 528, 555, 558, 597, 612, 615, 642

Offset: 1

Ray Chandler, Oct 26 2003

nonn

Numbers n such that 2*n-7 [A089192] and 2*n+7 [A105760] are both prime. [Vincenzo Librandi, Jul 10 2010]

Cf. A087697, A040040, A088763, A088765, A088769.

Cf. A089192, A105760. [Vincenzo Librandi, Jan 18 2009]

A154684 Triangle read by rows where T(m,n)=2mn + m + n - 3, 1<=n<=m.

Original entry on oeis.org

1, 4, 9, 7, 14, 21, 10, 19, 28, 37, 13, 24, 35, 46, 57, 16, 29, 42, 55, 68, 81, 19, 34, 49, 64, 79, 94, 109, 22, 39, 56, 73, 90, 107, 124, 141, 25, 44, 63, 82, 101, 120, 139, 158, 177, 28, 49, 70, 91, 112, 133, 154, 175, 196, 217, 31, 54, 77, 100, 123, 146, 169

Offset: 1

Vincenzo Librandi, Jan 18 2009

2*T(m,n)+7 = (2n+1)*(2m+1) is not prime.

First column: A016777; second column: A016897; third column: A008589; fourth column: A017173. - Vincenzo Librandi, Nov 19 2012

			Triangle begins:
1;
4,  9;
7,  14, 21;
10, 19, 28, 37;
13, 24, 35, 46, 57;
16, 29, 42, 55, 68,  81;
19, 34, 49, 64, 79,  94,  109;
22, 39, 56, 73, 90,  107, 124, 141;
25, 44, 63, 82, 101, 120, 139, 158, 177;
28, 49, 70, 91, 112, 133, 154, 175, 196, 217; etc.

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

Cf. A153053, A105760, A162265 (row sums), A016777, A016897, A008589, A017173.

[(2*n*k + n + k - 3): k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 19 2012

t[n_,k_]:=2 n*k + n + k - 3; Table[t[n, k], {n, 20}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 19 2012 *)

A023275 Primes that remain prime through 3 iterations of function f(x) = 2x + 7.

Original entry on oeis.org

5, 23, 293, 593, 953, 2063, 3323, 4133, 4583, 8243, 8783, 9173, 9203, 14723, 15383, 16103, 16763, 18413, 19163, 20123, 25733, 29453, 37223, 38783, 39443, 40253, 41903, 42923, 44753, 45863, 49433, 51473, 54443, 54623, 54713, 57383, 58913, 63353, 66533

Offset: 1

David W. Wilson

nonn

Primes p such that 2*p+7, 4*p+21 and 8*p+49 are also primes. - Vincenzo Librandi, Aug 04 2010

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

Subsequence of A023206, A023244, and of A105760.

[n: n in [1..100000] | IsPrime(n) and IsPrime(2*n+7) and IsPrime(4*n+21) and IsPrime(8*n+49)] // Vincenzo Librandi, Aug 04 2010

Select[Prime@ Range@ 7000, Times @@ Boole@ PrimeQ@ Rest@ NestList[2 # + 7 &, #, 3] > 0 &] (* Michael De Vlieger, Sep 19 2016 *)
Select[Prime[Range[7000]],AllTrue[Rest[NestList[2#+7&,#,3]],PrimeQ]&] (* Harvey P. Dale, Dec 26 2022 *)

is(n)=isprime(n) && isprime(2*n+7) && isprime(4*n+21) && isprime(8*n+49) \\ Charles R Greathouse IV, Sep 20 2016

a(n) == 23 (mod 30) for n > 1. - John Cerkan, Sep 16 2016

A023305 Primes that remain prime through 4 iterations of function f(x) = 2x + 7.

Original entry on oeis.org

293, 2063, 4583, 9203, 14723, 20123, 25733, 29453, 40253, 54713, 76103, 97523, 99833, 109433, 138683, 149993, 158243, 196853, 199403, 218873, 253103, 297623, 379913, 416963, 445463, 468113, 508073, 551963, 562403, 564713, 574703, 583733

Offset: 1

David W. Wilson

nonn

Primes p such that 2*p+7, 4*p+21, 8*p+49 and 16*p+105 are also primes. - Vincenzo Librandi, Aug 04 2010

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

Subsequence of A023206, A023244, A023275, and A105760.

[n: n in [1..1000000] | IsPrime(n) and IsPrime(2*n+7) and IsPrime(4*n+21) and IsPrime(8*n+49) and IsPrime(16*n+105)] // Vincenzo Librandi, Aug 04 2010

rp4Q[n_]:=AllTrue[Rest[NestList[2#+7&,n,4]],PrimeQ]; Select[Prime[Range[ 50000]],rp4Q] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 30 2020 *)

a(n) == 3 (mod 10). - John Cerkan, Oct 04 2016

A023333 Primes that remain prime through 5 iterations of function f(x) = 2x + 7.

Original entry on oeis.org

14723, 20123, 54713, 109433, 594653, 604883, 676493, 759953, 847103, 935843, 1035743, 1049603, 1079033, 1099823, 1222253, 1263323, 1499153, 1754033, 1835003, 1893173, 2017283, 2071493, 2099213, 2199653, 2895743, 2998313, 3389693, 4133663

Offset: 1

David W. Wilson

nonn

Primes p such that 2*p+7, 4*p+21, 8*p+49, 16*p+105 and 32*p+217 are also primes. - Vincenzo Librandi, Aug 04 2010

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

Subsequence of A023206, A023244, A023275, A023305, and A105760.

[n: n in [1..5000000] | IsPrime(n) and IsPrime(2*n+7) and IsPrime(4*n+21) and IsPrime(8*n+49) and IsPrime(16*n+105) and IsPrime(32*n+217)] // Vincenzo Librandi, Aug 04 2010

a(n) == 23 (mod 30). - John Cerkan, Oct 10 2016

A106086 Primes p such that 7p + 2 and 2p + 7 are primes.

Original entry on oeis.org

3, 5, 11, 23, 47, 53, 71, 131, 173, 197, 251, 257, 293, 317, 383, 461, 467, 587, 593, 683, 701, 773, 797, 863, 953, 983, 1031, 1103, 1151, 1187, 1193, 1217, 1301, 1307, 1373, 1451, 1481, 1607, 1721, 1787, 2111, 2207, 2237, 2333, 2633, 2903, 3023, 3221, 3347

Offset: 1

Zak Seidov, May 07 2005

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A105760 (2n+7 is prime), A105772 (7n+2 is prime).

Cf. A107438, A023206.

[p: p in PrimesUpTo(5000)|IsPrime(7*p+2) and IsPrime(2*p+7)] // Vincenzo Librandi, Jan 30 2011

Select[Prime[Range[220]], PrimeQ[2#+7]&&PrimeQ[7#+2]&]

More terms from Rick L. Shepherd, Jan 29 2006

A107438 Primes p such that 7p+2 or 2p+7 is prime.

Original entry on oeis.org

2, 3, 5, 11, 17, 23, 41, 47, 53, 71, 83, 101, 107, 113, 131, 137, 167, 173, 191, 197, 227, 251, 257, 281, 293, 311, 317, 347, 353, 383, 401, 431, 461, 467, 503, 521, 563, 587, 593, 641, 647, 677, 683, 701, 743, 773, 797, 821, 827, 857, 863, 887, 911, 941, 947

Offset: 1

Zak Seidov, May 26 2005

nonn

Cf. A105760 Numbers n such that (2*n + 7) is prime; A105772 Numbers n such that (7*n + 2) is prime.

Select[Prime[Range[220]], PrimeQ[2#+7]||PrimeQ[7#+2]&] (* Shepherd *)

isok(n) = isprime(n) && (isprime(7*n+2) || isprime(2*n+7)); \\ Michel Marcus, Oct 06 2013

Edited by Rick L. Shepherd, Feb 01 2006

cp's OEIS Frontend

A153053 Numbers j such that 2*j + 7 is not a prime.

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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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A023244 Primes that remain prime through 2 iterations of the function f(x) = 2x + 7.

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Mathematica

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A088767 a(n) = A087697(n)/2.

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A154684 Triangle read by rows where T(m,n)=2mn + m + n - 3, 1<=n<=m.

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Magma

Mathematica

A023275 Primes that remain prime through 3 iterations of function f(x) = 2x + 7.

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Magma

Mathematica

PARI

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A023305 Primes that remain prime through 4 iterations of function f(x) = 2x + 7.

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Magma

Mathematica

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A023333 Primes that remain prime through 5 iterations of function f(x) = 2x + 7.

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A106086 Primes p such that 7p + 2 and 2p + 7 are primes.

A107438 Primes p such that 7p+2 or 2p+7 is prime.