A067076 -id:A067076 - cp's OEIS frontend

A081759 Numbers n such that 5n+6 is prime.

1, 5, 7, 11, 13, 19, 25, 29, 35, 37, 41, 47, 49, 53, 55, 61, 65, 79, 83, 85, 91, 97, 103, 107, 113, 119, 125, 127, 131, 137, 139, 149, 151, 161, 163, 175, 181, 187, 193, 197, 203, 205, 209, 211, 217, 229, 233, 235, 239, 245, 257, 259, 263, 271, 275, 289

Offset: 1

Giovanni Teofilatto, Nov 21 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

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A024912, A030430, A049511, A067076, A087505, A089192, A089253.

[n: n in [0..300]| IsPrime(5*n + 6)]; // Vincenzo Librandi, Oct 16 2012

A081759 := proc(n) option remember: local k: if(n=1)then return 1: fi: for k from procname(n-1)+1 do if(isprime(5*k+6))then return k: fi: od: end: seq(A081759(n),n=1..100); # Nathaniel Johnston, May 28 2011

Select[Range[300], PrimeQ[5# + 6] &] (* Ray Chandler, Dec 06 2006 *)

is(n)=isprime(5*n+6) \\ Charles R Greathouse IV, Feb 17 2017

a(n) = 2*A024912(n) - 1.

Corrected by Ray Chandler, Nov 22 2003

A082749 Difference between the sum of next prime(n) natural numbers and the sum of next n primes.

Original entry on oeis.org

1, 4, 9, 10, 54, 71, 191, 236, 446, 1025, 1310, 2259, 3245, 3820, 5048, 7321, 10060, 11473, 15328, 18358, 20381, 25672, 30222, 36561, 46367, 53031, 58108, 65444, 70971, 78391, 104184, 116542, 133095, 142728, 169931, 181324, 203429, 226622

Offset: 1

Amarnath Murthy, Apr 17 2003

nonn

Group the natural numbers with prime(n) elements in each group. (1,2),(3,4,5),(6,7,8,9,10),(11,12,13,14,15,16,17),... The sum corresponding the groups is 3,12,40,98,... Group the prime numbers such that the n-th group contains n primes. (2),(3,5),(7,11,13),(17,19,23,29),... The sum corresponding the groups is 2,8,31,88,... The required difference is 1,4,9,10,...

The following sequences (allowing offset of first term) all appear to have the same parity: A034953, triangular numbers with prime indices; A054269, length of period of continued fraction for sqrt(p), p prime; A082749, difference between the sum of next prime(n) natural numbers and the sum of next n primes; A006254, numbers n such that 2n-1 is prime; A067076, 2n+3 is a prime. - Jeremy Gardiner, Sep 10 2004

Module[{nn=80,trms=40,c,nat,pr},c=(nn(nn+1))/2;nat=Total/@TakeList[Range[c],Prime[Range[trms]]];pr=Total/@TakeList[Prime[Range[c]], Range[trms]]; Differences/@ Thread[{pr,nat}]]//Flatten (* Harvey P. Dale, Apr 13 2025 *)

a(n) = ((A061802(n-1) + 1)*A000040(n))/2 - A007468(n). - Gionata Neri, May 17 2015

More terms from Ray Chandler, May 13 2003

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)

A047949 a(n) is the largest m such that n-m and n+m are both primes, or -1 if no such m exists.

Original entry on oeis.org

0, 0, 1, 2, 1, 4, 5, 4, 7, 8, 7, 10, 9, 8, 13, 14, 13, 12, 17, 16, 19, 20, 19, 22, 21, 20, 25, 24, 23, 28, 29, 28, 27, 32, 31, 34, 35, 34, 33, 38, 37, 40, 39, 38, 43, 42, 41, 30, 47, 46, 49, 50, 49, 52, 53, 52, 55, 54, 53, 48, 51, 50, 45, 62, 61, 64, 63, 62, 67, 68, 67, 66

Offset: 2

Lior Manor

A067076 is a subsequence of this sequence: when 2m+3 is prime a(m+3) = m. Moreover, it is the subsequence of records (maximal increasing subsequence): let m=a(n), with p=n-m and q=p+2m both odd primes > 3; now 3+2(m+(p-3)/2)=q and hence a(3+m+(p-3)/2) >= m+(p-3)/2 > m = a(n) but 3+m+(p-3)/2 < n. - Jason Kimberley, Aug 30 2012 and Oct 10 2012
Goldbach's conjecture says a(n) >= 0 for all n. - Robert Israel, Apr 15 2015
a(n) is the Goldbach partition of 2n which results in the maximum spread divided by 2. - Robert G. Wilson v, Jun 18 2018

			49-30=19 and 49+30=79 are primes, so a(49)=30.

T. D. Noe, Table of n, a(n) for n = 2..10000
OEIS (Plot 2), A067076 vs A098090 (n-m=3). - _Jason Kimberley_, Oct 01 2012

Cf. A047160, A067076, A182138.
Cf. A020481.
Cf. A010051.

a047949 n = if null qs then -1 else head qs  where
   qs = [m | m <- [n, n-1 .. 0], a010051' (n+m) == 1, a010051' (n-m) == 1]
-- Reinhard Zumkeller, Nov 02 2015

a:= proc(n)
local k;
  for k from n - 1 to 0 by -2 do
     if isprime(n+k) and isprime(n-k) then return(k) fi
od:
-1
end proc:
0, seq(a(n),n=3..1000); # Robert Israel, Apr 16 2015

a[2] = a[3] = 0; a[n_] := (For[m = n - 2, m >= 0, m--, If[PrimeQ[n - m] && PrimeQ[n + m], Break[]]]; m); Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Sep 04 2013 *)
lm[n_]:=Module[{m=n-2},While[!AllTrue[n+{m,-m},PrimeQ],m--];m]; Join[{0,0}, Array[ lm,70,4]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 03 2014 *)
f[n_] := Block[{q = 2}, While[q <= n && !PrimeQ[2n -q], q = NextPrime@ q]; n - q]; Array[f, 72, 2] (* Robert G. Wilson v, Jun 18 2018 *)

a(n) = {if (n==2 || n==3, return (0)); my(m = 1, lastm = -1, do = 1); while (do, if (isprime(n-m) && isprime(n+m), lastm = m); m++; if (m == n - 1, do = 0);); return (lastm);} \\ Michel Marcus, Jun 09 2013

a(n)=if(n<4,0,forprime(p=3,n-1,if(isprime(2*n-p),return(n-p)));-1) \\ Ralf Stephan, Dec 29 2013

a(n) = n - A020481(n).

a(n) = (A020482(n) - A020481(n))/2. - Gionata Neri, Apr 15 2015

Corrected by Harvey P. Dale, Dec 21 2000

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

A087915 Even numbers k such that 2*k+3 is a prime.

Original entry on oeis.org

0, 2, 4, 8, 10, 14, 20, 22, 28, 32, 34, 38, 40, 50, 52, 62, 64, 68, 74, 80, 82, 88, 94, 98, 104, 110, 112, 118, 124, 130, 134, 140, 152, 154, 164, 172, 178, 182, 188, 190, 208, 214, 218, 220, 230, 232, 238, 242, 244, 248, 250, 260, 272, 280, 284, 292, 298, 302

Offset: 1

Giovanni Teofilatto, Oct 18 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.

A subset of A067076.
Cf. A002145.
a(n) = A089193(n)-5.

Select[Range[0,350,2],PrimeQ[2#+3]&] (* Harvey P. Dale, Jan 12 2012 *)

More terms from Ray Chandler, Oct 19 2003

Offset corrected by Arkadiusz Wesolowski, Aug 09 2011

A078444 Floor of geometric mean of two consecutive primes.

Original entry on oeis.org

2, 3, 5, 8, 11, 14, 17, 20, 25, 29, 33, 38, 41, 44, 49, 55, 59, 63, 68, 71, 75, 80, 85, 92, 98, 101, 104, 107, 110, 119, 128, 133, 137, 143, 149, 153, 159, 164, 169, 175, 179, 185, 191, 194, 197, 204, 216, 224, 227, 230, 235, 239, 245, 253, 259, 265, 269, 273, 278

Offset: 1

Lior Manor, Dec 31 2002

For n > 1, a(n) = prime(n) iff prime(n) and prime(n+1) are twin primes.

			a(7) = floor(sqrt(prime(7)*prime(8))) = 17.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Andrica's Conjecture

Cf. A001223, A006094.

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

seq(floor(sqrt(ithprime(i)*ithprime(i+1))), i=1..100); # Robert Israel, Dec 12 2015

Table[Floor[Sqrt[Prime[n] Prime[n + 1]]], {n, 60}] (* Vincenzo Librandi, Dec 12 2015 *)
Table[Ceiling[(Prime[n] + Prime[n + 1])/2 - 1], {n, 100}] (* Miko Labalan, Dec 14 2015 *)

a(n) = sqrtint(prime(n)*prime(n+1)); \\ Michel Marcus, Dec 12 2015

a(n) = floor(sqrt(prime(n)*prime(n+1))).
From Miko Labalan, Dec 12 2015: (Start)
a(n) = A006254(A028310(n - 1)) + A067076(n);
a(n) = A067076(A028310(n - 1)) + A006254(n);
a(n) = A005097(A028310(n - 1)) + A005097(n).
(End)
For n >= 2 these formulas are equivalent to sqrt(prime(n)*prime(n+1)) > (prime(n)+prime(n+1))/2 - 1, and thus to A001223(n) <= 2 + 2*sqrt(2*prime(n)). This would be implied by Andrica's conjecture, but is as yet unproven. - Robert Israel, Dec 13 2015

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

A008507 Number of odd composite numbers less than n-th odd prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 5, 5, 7, 8, 8, 9, 11, 13, 13, 15, 16, 16, 18, 19, 21, 24, 25, 25, 26, 26, 27, 33, 34, 36, 36, 40, 40, 42, 44, 45, 47, 49, 49, 53, 53, 54, 54, 59, 64, 65, 65, 66, 68, 68, 72, 74, 76, 78, 78, 80, 81, 81, 85, 91, 92, 92, 93, 99, 101, 105, 105, 106, 108, 111, 113, 115, 116, 118

Offset: 1

Gary Findley (chfindley(AT)alpha.nlu.edu)

nonn

a(n) = A067076(n) - n + 1. - Vincenzo Librandi, Feb 02 2013

For n>=4, a(n) = k+1, where A000217(j) is the smallest triangular number such that A000217(j) - A033286(n+1) also is a triangular number, i.e., A000217(k). Example n=29, a(29) = 27: A033286(30) = 3390, A000217(86) = 3741. 3741-3390 = 351 = A000217(26); k=26, 26+1 = 27. - Bob Selcoe, Apr 12 2016

Cf. A067076.
Cf. A000040 (prime numbers), A000217 (triangular numbers), A033286 (n*prime(n)).
Partial sums of A100820.

[(Odd Primes - 1)/2] - n for n > 0, or A005097(n) - A000027(n). For example, A005097(1) - A000027(1) = 1 - 1 = 0, A005097(2) - A000027(2) = 2 - 2 = 0, A005097(9) - A000027(9) = 14 - 9 = 5. - William A. Tedeschi, Apr 25 2008

More terms from David W. Wilson, Jan 13 2000

A098033 Parity of p*(p+1)/2 for n-th prime p.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0

Offset: 1

Jeremy Gardiner, Sep 10 2004

The following sequences (possibly with a different offset for first term) all appear to have the same parity: A034953 = triangular numbers with prime indices; A054269 = length of period of continued fraction for sqrt(p), p prime; A082749 = difference between the sum of the next prime(n) natural numbers and the sum of the next n primes; A006254 = numbers n such that 2n-1 is prime; A067076 = numbers n such that 2n+3 is a prime.
Analogous to the prime race (mod 3). - Robert G. Wilson v, Sep 17 2004
See also A089253 = 2n-5 is a prime.
For n > 1, if A000040(n) == 1 (mod 4), then a(n) = 1, otherwise a(n)=0, so (for n>1) also a(n) = number of representations of A000040(n) as a difference of hexagonal numbers (A000384) (cf. [Nyblom, p. 262]). - L. Edson Jeffery, Feb 16 2013

			a(1) = parity of (2*(2+1)/2 = 3) = 1 (odd).

Amiram Eldar, Table of n, a(n) for n = 1..10000
M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.

Cf. A034953, A054269, A082749, A006254, A067076, A100672.

seq((ithprime(n) mod 4) mod 3, n = 2..105] # Gary Detlefs, Oct 27 2011

Table[ Mod[ Prime[n](Prime[n] + 1)/2, 2], {n, 105}] (* Robert G. Wilson v, Sep 17 2004 *)
Mod[(#(#+1))/2,2]&/@Prime[Range[110]] (* Harvey P. Dale, Mar 29 2015 *)

a(n)=prime(n)%4<3 \\ Charles R Greathouse IV, Oct 27 2011

a(n) = parity of p*(p+1)/2 for n-th prime p.
a(n) = 1 - A100672(n), n > 1. - Steven G. Johnson (stevenj(AT)math.mit.edu), Sep 18 2008
For n > 1, a(n) = (prime(n) mod 4) mod 3. - Gary Detlefs, Oct 27 2011

More terms from Robert G. Wilson v, Sep 17 2004

cp's OEIS Frontend

A081759 Numbers n such that 5n+6 is prime.

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A082749 Difference between the sum of next prime(n) natural numbers and the sum of next n primes.

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

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A047949 a(n) is the largest m such that n-m and n+m are both primes, or -1 if no such m exists.

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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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A087915 Even numbers k such that 2*k+3 is a prime.

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A078444 Floor of geometric mean of two consecutive primes.

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