A124518 -id:A124518 - cp's OEIS frontend

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

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

A045753 Numbers n such that 4n-1 and 4n+1 are both primes.

Original entry on oeis.org

1, 3, 15, 18, 27, 45, 48, 57, 60, 78, 87, 105, 108, 150, 165, 207, 255, 258, 273, 288, 330, 357, 363, 372, 402, 405, 417, 447, 468, 483, 507, 522, 528, 567, 585, 648, 672, 678, 750, 780, 792, 813, 825, 840, 843, 867, 882, 885, 918, 942, 963, 1005, 1023

Offset: 1

Felice Russo

			3 belongs to the sequence because 4*3+1 and 4*3-1 are both primes.

Zak Seidov, Table of n, a(n) for n = 1..10000

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

[n: n in [1..2000] | IsPrime(4*n+1) and IsPrime(4*n-1)] // Vincenzo Librandi, Nov 18 2010

Select[Range[1023], And @@ PrimeQ[{-1, 1} + 4# ] &] (* Ray Chandler, Dec 06 2006 *)

list(lim)=my(v=List(),p=2); forprime(q=3,4*lim+1, if(q-p==2 && p%4==3, listput(v,q\4)); p=q); Vec(v) \\ Charles R Greathouse IV, Dec 03 2016

More terms from Erich Friedman

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

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

A058218 Positive integers that cannot be represented in the form n=5|ab|+a+b for any choice of nonzero integers a and b (positive or negative).

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 14, 18, 22, 26, 28, 30, 32, 36, 40, 44, 48, 50, 54, 58, 60, 62, 66, 76, 78, 82, 84, 94, 96, 98, 100, 102, 104, 114, 116, 120, 126, 132, 136, 138, 140, 144, 150, 154, 158, 162, 166, 170, 176, 184, 188, 190, 198, 202, 204, 208, 210, 212, 216, 220

Offset: 1

John W. Layman, Nov 30 2000

nonn

All terms except 1 are even. - Robert Israel, Apr 07 2019

Robert Israel, Table of n, a(n) for n = 1..10000
Maria Suzuki, Related to Alternative Formulations of the Twin Prime Problem, American Math. Monthly, 107 (2000) pp. 55-56.

A002822 results if the coefficient 5 in the definition above is replaced by 6.

Includes 2*A124518.

filter:= proc(n)
   nops(select(t -> t mod 5 = 1 or t mod 5 = 4, numtheory:-divisors(5*n+1))) = 2
   and nops(select(t -> t mod 5 = 4, numtheory:-divisors(5*n-1)))=1
end proc:
select(filter, [$1..1000]); # Robert Israel, Apr 07 2019

filterQ[n_] := Length[Select[Divisors[5 n + 1], Mod[#, 5] == 1 || Mod[#, 5] == 4&]] == 2 && Length[Select[Divisors[5 n - 1], Mod[#, 5] == 4&]] == 1;
Select[Range[1000], filterQ] (* Jean-François Alcover, Aug 16 2020, after Robert Israel *)

A303550 Numbers k such that abs(60k^2 - 1710k + 12150) +- 1 are twin primes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 27, 33, 34, 35, 36, 38, 41, 50, 56, 57, 64, 66, 69, 75, 81, 85, 86, 90, 93, 98, 103, 106, 119, 121, 133, 136, 141, 143, 146, 150, 181, 182, 189, 195, 202, 207, 208, 212, 215, 218, 219, 225

Offset: 1

Amiram Eldar, Apr 26 2018

The formula was discovered by Andrew T. Gazsi in 1961.
The polynomial can also be given as 30*(2*k - 27)*(k - 15). Its value is negative (-30) at k = 14 and 0 and k = 15.
Beiler erroneously claimed that the polynomial generates twin primes for k = 1 to 20.

			1 is in the sequence since 60*1^2 - 1710*1 + 12150 = 10500 and (10499, 10501) are twin primes.

Albert H. Beiler, Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, 2nd ed., Dover Publications, Inc., New York, 1966, p. 225.
Joseph B. Dence and Thomas P. Dence, Elements of the Theory of Numbers, Academic Press, 1999, problem 1.94, p.35.
Andrew T. Gazsi, A Formula to Generate Prime Pairs, Recreational Mathematics Magazine, edited by Joseph S. Madachy, Issue 6, December 1961, p. 44.

Giovanni Resta, Table of n, a(n) for n = 1..10000
James Alston Hope Hunter and Joseph S. Madachy, Mathematical Diversions, D. van Nostrand Company, Inc., Princeton, New Jersey, 1963, p. 7.
Carlos Rivera, Problem 44. Twin-primes producing polynomials race, The Prime Puzzles & Problems Connection.

Cf. A001097, A001359, A006512, A088485, A108897, A124518, A124519, A139404, A185660.

filter:= proc(n) local k;
  k:= abs(60*n^2-1710*n+12150);
  isprime(k+1) and isprime(k-1)
end proc:
select(filter, [$1..300]); # Robert Israel, Jun 19 2018

f[n_] := 60n^2 - 1710n + 12150; aQ[n_]:=PrimeQ[f[n]-1] && PrimeQ[f[n]+1]; Select[Range[225], aQ]
Select[Range[250],AllTrue[Abs[60#^2-1710#+12150+{1,-1}],PrimeQ]&] (* Harvey P. Dale, May 17 2025 *)

f(n) = abs(60*n^2 - 1710*n + 12150);
isok(n) = my(fn=f(n)); isprime(fn-1) && isprime(fn+1); \\ Michel Marcus, Apr 27 2018

cp's OEIS Frontend

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

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A045753 Numbers n such that 4n-1 and 4n+1 are both 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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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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A124521 Numbers k such that 16*k - 1 and 16*k + 1 are twin primes.

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Maple

Mathematica

Extensions

A058218 Positive integers that cannot be represented in the form n=5|ab|+a+b for any choice of nonzero integers a and b (positive or negative).

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.

A303550 Numbers k such that abs(60k^2 - 1710k + 12150) +- 1 are twin primes.