A260689 -id:A260689 - cp's OEIS frontend

A040040 Average of twin prime pairs (A014574), divided by 2. Equivalently, 2a(n)-1 and 2a(n)+1 are primes.

2, 3, 6, 9, 15, 21, 30, 36, 51, 54, 69, 75, 90, 96, 99, 114, 120, 135, 141, 156, 174, 210, 216, 231, 261, 285, 300, 309, 321, 330, 405, 411, 414, 429, 441, 510, 516, 525, 531, 546, 576, 615, 639, 645, 651, 660, 714, 726, 741, 744, 804, 810, 834, 849, 861, 894

Offset: 1

N. J. A. Sloane

Intersection of A005097 and A006254. - Zak Seidov, Mar 18 2005
The only possible pairs for 2a(n)+-1 are prime/prime (this sequence), not prime/not prime (A104278), prime/notprime (A104279) and not prime/prime (A104280), ... this sequence + A104280 + A104279 + A104278 = the odd numbers.
These numbers are never k mod (2k+1) or (k+1) mod (2k+1) with 2k+1 < a(n). - Jon Perry, Sep 04 2012
Excluding the first term, all remaining terms have digital root 3, 6 or 9. - J. W. Helkenberg, Jul 24 2013
Positive numbers x such that the difference between x^2 and adjacent squares are prime (both x^2-(x-1)^2 and (x+1)^2-x^2 are prime). - Doug Bell, Aug 21 2015

T. D. Noe, Table of n, a(n) for n = 1..10001

Cf. A001359, A006512, A014574, A054735, A111046, A045753 (even terms halved), A002822 (terms divided by 3).
Cf. A221310.
Cf. A260689, A264526.

a040040 = flip div 2 . a014574  -- Reinhard Zumkeller, Nov 17 2015

P := select(isprime,[$1..1789]): map(p->(p+1)/2, select(p->member(p+2,P),P)); # Peter Luschny, Mar 03 2011

Select[Range[900], And @@ PrimeQ[{-1, 1} + 2# ] &] (* Ray Chandler, Oct 12 2005 *)

p=2; forprime(b=3, 1e4, if(b-p==2, print1((p+1)/2", ")); p=b) \\ Altug Alkan, Nov 10 2015

a(n) = A014574(n)/2 = A054735(n+1)/4 = A111046(n+1)/8.
For n > 1, a(n) = 3*A002822(n-1). - Jason Kimberley, Nov 06 2015
A260689(a(n),1) = A264526(a(n)) = 1. - Reinhard Zumkeller, Nov 17 2015
From Michael G. Kaarhus, Aug 19 2022: (Start)
a(n) = (A001359(n) + 1)/2.
a(n) = (A006512(n) - 1)/2.
For n > 1, a(n) = A167379(n-1) * 3/2. (End)

More terms from Cino Hilliard, Oct 21 2002
Title corrected by Daniel Forgues, Jun 01 2009
Edited by Daniel Forgues, Jun 21 2009
Comment corrected by Daniel Forgues, Jul 12 2009

A069360 Number of prime pairs (p,q), p <= q, such that (p+q)/2 = 2*n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 2, 2, 4, 3, 3, 5, 3, 3, 6, 5, 2, 6, 5, 4, 8, 4, 4, 7, 6, 5, 8, 7, 6, 12, 5, 3, 9, 5, 7, 11, 5, 4, 11, 8, 5, 13, 6, 7, 14, 8, 5, 11, 9, 8, 14, 7, 6, 13, 9, 7, 12, 7, 9, 18, 9, 6, 16, 8, 10, 16, 9, 7, 16, 14, 8, 17, 8, 8, 21, 10, 8, 17, 10, 11

Offset: 1

Reinhard Zumkeller, Apr 15 2002

The Goldbach conjecture, if true, would imply a(n) > 0.

Row lengths of table A260689, n > 1. - Reinhard Zumkeller, Nov 17 2015

			n=8: there are 16 pairs (i,j) with (i+j)/2=n*2=16; only two of them, (3,29) and (13,19), consist of primes, therefore a(8)=2.

Klaus Brockhaus, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Goldbach conjecture

Bisection of A002375.
Cf. A082467 (least k such that n-k and n+k are both primes), A134677 (records), A134678 (where records occur), A135146 (index of first occurrence of n).
Cf. A000040, A010051, A260689.

a069360 n = sum [a010051' (4*n-p) | p <- takeWhile (<= 2*n) a000040_list]
-- Reinhard Zumkeller, May 08 2014, Apr 09 2012

Table[Length[Select[Range[0, 2*n], PrimeQ[2n-#] && PrimeQ[2n+#] &]], {n, 50}] (* Stefan Steinerberger, Nov 30 2007 *)
Table[Boole[n == 1] + Sum[Boole[PrimeQ@ i] Boole[PrimeQ[4 n - i]] Mod[Sign[4 n - i], 4], {i, 3, 2 n}], {n, 80}] (* Michael De Vlieger, Dec 21 2016 *)
Table[Count[IntegerPartitions[4n,{2}],?(AllTrue[#,PrimeQ]&)],{n,80}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Mar 09 2018 *)

a(n)=my(s);forprime(p=2,2*n,s+=isprime(4*n-p));s \\ Charles R Greathouse IV, Apr 09 2012

For n > 1: a(n) = #{k | 2*n-k and 2*n+k are prime, 1<=k<=2*n}.

a(n) = Sum_{i=3..2n} isprime(i) * isprime(4n-i) * (sign(4n-i) mod 4), n > 1. - Wesley Ivan Hurt, Dec 18 2016

Edited by Klaus Brockhaus, Nov 20 2007

a(1)=1, thanks to Charles R Greathouse IV, who noticed this; b-file adjusted.

A088763 a(n) = A087695(n)/2.

Original entry on oeis.org

4, 5, 7, 8, 10, 13, 17, 20, 22, 25, 28, 32, 35, 38, 43, 50, 52, 53, 55, 67, 77, 80, 85, 88, 97, 98, 113, 115, 118, 127, 130, 133, 137, 140, 155, 157, 167, 175, 178, 185, 188, 193, 218, 223, 230, 232, 253, 272, 280, 283, 287, 295, 298, 302, 305, 308, 322, 325, 328, 340

Offset: 1

Ray Chandler, Oct 26 2003

nonn

A260689(a(n),1) = A264526(a(n)) = 3. - Reinhard Zumkeller, Nov 17 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A087695, A040040, A088765, A088767, A088769.

Cf. A260689, A264526.

a088763 = flip div 2 . a087695  -- Reinhard Zumkeller, Nov 17 2015

ZL:=[]:for p from 1 to 700 do if (isprime(p) and isprime(p+6) ) then ZL:=[op(ZL),(p+(p+6))/4]; fi; od; print(ZL); # Zerinvary Lajos, Mar 07 2007

f[n_]:=PrimeQ[n-3]&&PrimeQ[n+3]; lst={};Do[If[f[n],AppendTo[lst,n]],{n,2,8!,2}];lst/2 (* Vladimir Joseph Stephan Orlovsky, Oct 09 2009 *)

Offset corrected by Reinhard Zumkeller, Nov 17 2015

A264526 Smallest number m such that both 2n-m and 2n+m are primes.

Original entry on oeis.org

1, 1, 3, 3, 1, 3, 3, 1, 3, 9, 5, 3, 9, 1, 9, 3, 5, 9, 3, 1, 3, 15, 5, 3, 9, 7, 3, 15, 1, 9, 3, 5, 15, 3, 1, 15, 3, 5, 9, 15, 5, 3, 9, 7, 9, 15, 7, 9, 3, 1, 3, 3, 1, 3, 15, 13, 15, 9, 7, 9, 15, 13, 21, 21, 5, 3, 27, 1, 9, 15, 5, 33, 9, 1, 15, 3, 7, 9, 3, 5

Offset: 2

Reinhard Zumkeller, Nov 17 2015

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A014574, A040040, A082467, A088763, A260689, A264527.

Haskell
```
a264526 = head . a260689_row
```

snm[n_]:=Module[{m=1},While[!PrimeQ[2n-m]||!PrimeQ[2n+m],m=m+2];m]; Array[ snm,90,2] (* Harvey P. Dale, Aug 13 2017, optimized by Ivan N. Ianakiev, Mar 16 2018 *)

a(n) = {my(m=1); while(!(isprime(2*n-m) && isprime(2*n+m)), m+=2); m;} \\ Michel Marcus, Mar 18 2018

a(n) = A260689(n,1);
a(A040040(n)) = 1;
a(A014574(n)/2) = 1;
a(A088763(n)) = 3.
a(n) = A082467(2n). - Ivan N. Ianakiev, Oct 27 2021

A264527 Largest number m such that (2n-m, 2n+m) is a prime pair.

Original entry on oeis.org

1, 1, 5, 7, 7, 9, 13, 13, 17, 19, 19, 21, 25, 23, 29, 27, 31, 35, 33, 37, 39, 43, 41, 47, 49, 49, 53, 55, 53, 51, 45, 61, 63, 67, 67, 63, 73, 73, 77, 75, 79, 81, 85, 83, 89, 87, 85, 95, 97, 97, 93, 93, 103, 87, 99, 109, 113, 115, 113, 119, 117, 115, 123, 127

Offset: 2

Reinhard Zumkeller, Nov 17 2015

nonn

a(n) = A260689(n,A069360(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A069360, A260689, A264526.

Haskell
```
a264527 = last . a260689_row
```

cp's OEIS Frontend

A040040 Average of twin prime pairs (A014574), divided by 2. Equivalently, 2*a(n)-1 and 2*a(n)+1 are primes.

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A069360 Number of prime pairs (p,q), p <= q, such that (p+q)/2 = 2*n.

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A088763 a(n) = A087695(n)/2.

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Maple

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A264526 Smallest number m such that both 2*n-m and 2*n+m are primes.

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Programs

Haskell

Mathematica

PARI

Formula

A264527 Largest number m such that (2*n-m, 2*n+m) is a prime pair.

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Author

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Programs

Haskell

A040040 Average of twin prime pairs (A014574), divided by 2. Equivalently, 2a(n)-1 and 2a(n)+1 are primes.

A264526 Smallest number m such that both 2n-m and 2n+m are primes.

A264527 Largest number m such that (2n-m, 2n+m) is a prime pair.