A045753 -id:A045753 - 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

A063983 Least k such that k*2^n +/- 1 are twin primes.

Original entry on oeis.org

4, 2, 1, 9, 12, 6, 3, 9, 57, 30, 15, 99, 165, 90, 45, 24, 12, 6, 3, 69, 132, 66, 33, 486, 243, 324, 162, 81, 90, 45, 345, 681, 585, 375, 267, 426, 213, 429, 288, 144, 72, 36, 18, 9, 147, 810, 405, 354, 177, 1854, 927, 1125, 1197, 666, 333, 519, 1032, 516, 258, 129, 72

Offset: 0

Robert G. Wilson v, Sep 06 2001

nonn

Excluding the first three terms, all remaining terms have digital root 3, 6, or 9. - J. W. Helkenberg, Jul 24 2013

			a(3) = 9 because 9*2^3 = 72 and 71 and 73 are twin primes.
a(6) = 3 because 3*2^6 = 192 and {191, 193} are twin primes.
a(71) = 630 because 630*2^71 = 1487545442103938242314240 and {1487545442103938242314239, 1487545442103938242314241} are twin primes.

Richard Crandall and Carl Pomerance, 'Prime Numbers: A Computational Perspective,' Springer-Verlag, NY, 2001, page 12.

Pierre CAMI, Table of n, a(n) for n = 0..2300

Cf. A040040, A045753, A002822, A124065, A124518-A124522.
Cf. A071256, A060210, A060256. For records see A125848, A125019.
Cf. A076806 (requires odd k).

Table[Do[s=(2^j)*k; If[PrimeQ[s-1]&&PrimeQ[s+1],Print[{j,k}]], {k,1,2*j^2}],{j,0,100}]; (* outprint of a[j]=k *)
Do[ k = 1; While[ ! PrimeQ[ k*2^n + 1 ] || ! PrimeQ[ k*2^n - 1 ], k++ ]; Print[ k ], {n, 0, 50} ]
f[n_] := Block[{k = 1},While[Nand @@ PrimeQ[{-1, 1} + 2^n*k], k++ ];k];Table[f[n], {n, 0, 60}] (* Ray Chandler, Jan 09 2009 *)

More terms from Labos Elemer, May 24 2002

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

A124519 Numbers k such that 12k - 1 and 12k + 1 are twin primes.

Original entry on oeis.org

1, 5, 6, 9, 15, 16, 19, 20, 26, 29, 35, 36, 50, 55, 69, 85, 86, 91, 96, 110, 119, 121, 124, 134, 135, 139, 149, 156, 161, 169, 174, 176, 189, 195, 216, 224, 226, 250, 260, 264, 271, 275, 280, 281, 289, 294, 295, 306, 314, 321, 335, 341, 344, 355, 356, 379, 399

Offset: 1

Artur Jasinski, Nov 04 2006

nonn

			1 is in the sequence since 12*1 - 1 = 11 and 12*1 + 1 = 13 are twin primes.

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

Cf. A040040, A045753, A002822, A124065, A124518, A124520, A124521, A124522, A063983.

Select[Range[400], And @@ PrimeQ[{-1, 1} + 12# ] &] (* Ray Chandler, Nov 16 2006 *)

Extended by Ray Chandler, Nov 16 2006

A124522 a(n) = smallest k such that 2nk-1 and 2nk+1 are primes.

Original entry on oeis.org

2, 1, 1, 9, 3, 1, 3, 12, 1, 3, 9, 3, 12, 15, 1, 6, 3, 2, 6, 6, 1, 15, 3, 4, 3, 6, 2, 48, 6, 1, 21, 3, 3, 15, 6, 1, 27, 3, 4, 3, 15, 5, 12, 15, 2, 9, 3, 2, 9, 6, 1, 3, 60, 1, 6, 24, 2, 3, 9, 2, 129, 12, 7, 9, 15, 5, 12, 27, 1, 3, 9, 3, 42, 45, 1, 90, 3, 2, 66, 21, 5, 63, 27, 16, 6, 6, 2, 12, 24, 1, 6

Offset: 1

Artur Jasinski, Nov 04 2006

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A040040, A045753, A002822, A124065, A124518-A124522, A063983.

isA001359 := proc(n) RETURN( isprime(n) and isprime(n+2)) ; end: A124522 := proc(n) local k; k :=1 ; while true do if isA001359(2*n*k-1) then RETURN(k) ; fi ; k := k+1 ; od ; end: for n from 1 to 60 do printf("%d,",A124522(n)) ; od ; # R. J. Mathar, Nov 06 2006

f[n_] := Block[{k = 1},While[Nand @@ PrimeQ[{-1, 1} + 2n*k], k++ ];k];Table[f[n], {n, 91}] (* Ray Chandler, Nov 16 2006 *)
skp[n_]:=Module[{k=1},While[AnyTrue[2n k+{1,-1},CompositeQ],k++];k]; Join[{2},Array[skp,100,2]] (* Harvey P. Dale, Mar 30 2024 *)

{for(n=1,91,k=1;while(!isprime(2*n*k-1)||!isprime(2*n*k+1),k++);print1(k, ","))}

Edited and extended by Klaus Brockhaus and R. J. Mathar, Nov 06 2006

A124518 Numbers k such that 10k-1 and 10k+1 are twin primes.

Original entry on oeis.org

3, 6, 15, 18, 24, 27, 42, 57, 60, 66, 81, 102, 105, 123, 129, 132, 162, 195, 213, 231, 234, 255, 273, 279, 297, 300, 312, 330, 333, 336, 339, 354, 393, 402, 405, 423, 426, 465, 480, 501, 510, 528, 552, 564, 585, 588, 609, 627, 630, 636, 645, 657, 666, 669, 678

Offset: 1

Artur Jasinski, Nov 04 2006

nonn

All terms are divisible by 3. - Robert Israel, Apr 07 2019

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A040040, A045753, A002822, A124065, A124519-A124522, A063983.

select(t -> isprime(10*t+1) and isprime(10*t-1), [seq(i,i=3..1000,3)]); # Robert Israel, Apr 07 2019

Select[Range[678], And @@ PrimeQ[{-1, 1} + 10# ] &] (* Ray Chandler, Nov 16 2006 *)

A124065 Numbers k such that 8k - 1 and 8k + 1 are twin primes.

Original entry on oeis.org

9, 24, 30, 39, 54, 75, 129, 144, 165, 186, 201, 234, 261, 264, 324, 336, 339, 375, 390, 396, 420, 441, 459, 471, 516, 534, 600, 621, 654, 660, 690, 705, 735, 795, 819, 849, 870, 891, 936, 945, 1011, 1029, 1125, 1155, 1179, 1215, 1221, 1251, 1284, 1395, 1419

Offset: 1

Artur Jasinski, Nov 04 2006

nonn

			9 is in the sequence since 8*9 - 1 = 71 and 8*9 + 1 = 73 are twin primes.

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

Intersection of A005122 and A005123.

Cf. A040040, A045753, A002822, A124518, A124519, A124520, A124521, A124522, A063983.

[n: n in [1..2000] | IsPrime(8*n+1) and IsPrime(8*n-1)] // Vincenzo Librandi, Mar 08 2010

Select[Range[1500], And @@ PrimeQ[{-1, 1} + 8# ] &] (* Ray Chandler, Nov 16 2006 *)

from sympy import isprime
def ok(n): return isprime(8*n - 1) and isprime(8*n + 1)
print(list(filter(ok, range(1420)))) # Michael S. Branicky, Sep 24 2021

Extended by Ray Chandler, Nov 16 2006

A124520 Numbers k such that 14k - 1 and 14k + 1 are twin primes.

Original entry on oeis.org

3, 30, 33, 63, 75, 78, 93, 102, 123, 138, 153, 162, 165, 192, 195, 240, 252, 273, 297, 303, 342, 387, 393, 420, 435, 438, 450, 468, 483, 522, 525, 540, 588, 630, 633, 660, 663, 717, 738, 747, 750, 765, 798, 825, 837, 855, 957, 978, 993, 1023, 1032, 1062

Offset: 1

Artur Jasinski, Nov 04 2006

nonn

			3 is in the sequence since 14*3 - 1 = 41 and 14*3 + 1 = 43 are twin primes.

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

Cf. A040040, A045753, A002822, A124065, A124518, A124519, A124521, A124522, A063983.

Select[Range[1100], And @@ PrimeQ[{-1, 1} + 14# ] &] (* Ray Chandler, Nov 16 2006 *)

Extended by Ray Chandler, Nov 16 2006

A124521 Numbers k such that 16k - 1 and 16k + 1 are twin primes.

Original entry on oeis.org

12, 15, 27, 72, 93, 117, 132, 162, 168, 195, 198, 210, 258, 267, 300, 327, 330, 345, 435, 468, 642, 765, 813, 855, 903, 912, 960, 978, 993, 1128, 1143, 1182, 1290, 1350, 1353, 1365, 1392, 1398, 1440, 1632, 1680, 1713, 1737, 1797, 1848, 1860, 1947, 1953, 1962

Offset: 1

Artur Jasinski, Nov 04 2006

nonn

			12 is in the sequence since 16*12 - 1 = 191 and 16*12 + 1 = 193 are twin primes.

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

Cf. A040040, A045753, A002822, A124065, A124518, A124519, A124520, A124522, A063983.

A124521:=n->`if`(isprime(16*n-1) and isprime(16*n+1), n, NULL): seq(A124521(n), n=1..2000); # Wesley Ivan Hurt, Oct 10 2014

Select[Range[2000], And @@ PrimeQ[{-1, 1} + 16# ] &] (* Ray Chandler, Nov 16 2006 *)

Extended by Ray Chandler, Nov 16 2006

A365411 Numbers k such that 4k-1 and 4k+1 are both prime powers (A246655).

Original entry on oeis.org

1, 2, 3, 6, 7, 12, 15, 18, 20, 27, 42, 45, 48, 57, 60, 78, 87, 90, 105, 108, 150, 165, 182, 207, 210, 255, 258, 273, 288, 330, 342, 357, 363, 372, 402, 405, 417, 447, 462, 468, 483, 507, 522, 528, 552, 567, 585, 600, 648, 672, 678, 750, 780, 792, 813, 825, 840, 843

Offset: 1

Jianing Song, Oct 22 2023

Let b(q) be the number of pairs of consecutive nonzero squares in the finite field F_q for odd prime powers q, then b(q) = b(q') for q < q' if and only if q = 4*k-1 and q' = 4*k+1 for k being a term of this sequence, in which case we have b(q) = b(q') = k-1.

			6 is a term since 4*6-1 = 23 is a prime, and 4*6+1 = 25 is a prime power.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A246655, A366502. Supersequence of A045753.

{2*a(n)} is a subsequence of A365416.

isA365411(n) = isprimepower(4*n-1) && isprimepower(4*n+1)

A123200 Numbers k such that 1000000k-1 and 1000000k+1 are twin primes.

Original entry on oeis.org

24, 30, 198, 345, 348, 432, 438, 471, 492, 609, 669, 774, 777, 858, 864, 1032, 1083, 1125, 1218, 1395, 1536, 1824, 1914, 1929, 2088, 2139, 2301, 2334, 2376, 2418, 2448, 2460, 2544, 2763, 2832, 2970, 3021, 3297, 3369, 3384, 3495, 3528, 3540, 3633, 3777

Offset: 1

Jonathan Vos Post, Nov 05 2006

			a(6) = 432 because 431999999 and 432000001 are primes.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A000040, A040040, A045753, A002822, A124065.

a:=proc(n) if isprime(10^6*n-1)=true and isprime(10^6*n+1)=true then n else fi end: seq(a(n),n=1..4500); # Emeric Deutsch, Nov 16 2006

Select[Range[3800],AllTrue[#*10^6+{1,-1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 13 2017 *)

More terms from Emeric Deutsch, Nov 16 2006

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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A063983 Least k such that k*2^n +/- 1 are twin primes.

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A124519 Numbers k such that 12*k - 1 and 12*k + 1 are twin primes.

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A124522 a(n) = smallest k such that 2nk-1 and 2nk+1 are primes.

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A124518 Numbers k such that 10k-1 and 10k+1 are twin primes.

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A124065 Numbers k such that 8*k - 1 and 8*k + 1 are twin primes.

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A124520 Numbers k such that 14*k - 1 and 14*k + 1 are twin primes.

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

A124519 Numbers k such that 12k - 1 and 12k + 1 are twin primes.

A124065 Numbers k such that 8k - 1 and 8k + 1 are twin primes.

A124520 Numbers k such that 14k - 1 and 14k + 1 are twin primes.

A124521 Numbers k such that 16k - 1 and 16k + 1 are twin primes.

A365411 Numbers k such that 4k-1 and 4k+1 are both prime powers (A246655).

A123200 Numbers k such that 1000000k-1 and 1000000k+1 are twin primes.