A040040 -id:A040040 - cp's OEIS frontend

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

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

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

A088765 a(n) = A087696(n)/2.

Original entry on oeis.org

4, 6, 9, 12, 18, 21, 24, 33, 39, 42, 51, 54, 66, 72, 81, 84, 93, 114, 117, 123, 138, 144, 156, 171, 177, 189, 192, 207, 213, 219, 222, 231, 252, 276, 291, 306, 318, 324, 339, 348, 357, 369, 378, 396, 408, 417, 429, 441, 462, 471, 486, 507, 513, 522, 528, 546

Offset: 1

Ray Chandler, Oct 26 2003

nonn

"Example 3: Ordinary twins 2a +- 1 for a = 2, 3, 6, 9, 15, . . . have D = 1 and are in class I. For D = 3, the twins 2a +- 3 occur for a = 4, 5, 7, 8, 10"; the latter is this sequence, from p. 3 of Weber. - Jonathan Vos Post, Feb 14 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..3100
H. J. Weber, Exceptional Prime Number Twins, Triplets and Multiplets, arXiv:1102.3075 [math.NT], 2011.

Cf. A087696, A040040, A088763, A088767, A088769.

[n/2: n in [3..2000] |IsPrime(n+5) and IsPrime(n-5)]; // Vincenzo Librandi, May 20 2017

Select[Range[3, 2000], PrimeQ[# + 5] && PrimeQ[# - 5] &] / 2 (* Vincenzo Librandi, May 20 2017 *)

Offset changed from 0 to 1 by Vincenzo Librandi, May 21 2017

A104278 Numbers m such that 2m+1 and 2m-1 are not primes.

Original entry on oeis.org

13, 17, 25, 28, 32, 38, 43, 46, 47, 58, 59, 60, 61, 62, 67, 71, 72, 73, 77, 80, 85, 88, 92, 93, 94, 101, 102, 103, 104, 107, 108, 109, 110, 118, 122, 123, 124, 127, 130, 133, 137, 143, 144, 145, 148, 149, 150, 151, 152, 160, 161, 162, 163, 164, 167, 170, 171, 172

Offset: 1

Alexandre Wajnberg, Apr 17 2005

Complement of A147820. - Omar E. Pol, Nov 17 2008

m is in the sequence iff A177961(m)Vladimir Shevelev, May 16 2010

			a(1)=13 is the first number satisfying simultaneously the two rules.

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

Intersection of A047845 and A104275.
Cf. A040040, A147820, A025583.
Cf. A010051, A002808, A099047.

a104278 n = a104278_list !! (n-1)
a104278_list = [m | m <- [1..],
                    a010051' (2 * m - 1) == 0 && a010051' (2 * m + 1) == 0]
-- Reinhard Zumkeller, Aug 04 2015

Select[ Range[300], !PrimeQ[2# + 1] && !PrimeQ[2# - 1] &] (* Robert G. Wilson v, Apr 18 2005 *)
Select[Range[300],NoneTrue[2#+{1,-1},PrimeQ]&] (* The program uses the NoneTrue function from Mathematica version 10 *)  (* Harvey P. Dale, Jul 07 2015 *)

select( {is_A104278(n)=!isprime(2*n-1)&&!isprime(2*n+1)}, [1..222]) \\ M. F. Hasler, Apr 29 2024

a(n) = (A025583-1)/2. - Bill McEachen, Feb 05 2025

More terms from Robert G. Wilson v, Apr 18 2005

A066466 Numbers having just one anti-divisor.

Original entry on oeis.org

3, 4, 6, 96, 393216

Offset: 1

Robert G. Wilson v, Jan 02 2002

See A066272 for definition of anti-divisor.
Jon Perry calls these anti-primes.
A066272(a(n)) = 1.
From Max Alekseyev, Jul 23 2007; updated Jun 25 2025: (Start)
Except for a(2) = 4, the terms of A066466 have form 2^k*p where p is odd prime and both 2^(k+1)*p-1, 2^(k+1)*p+1 are prime (i.e., twin primes). In other words, this sequence, omitting 4, is a subsequence of A040040 containing elements of the form 2^k*p with prime p.
Furthermore, since 2^(k+1)*p-1, 2^(k+1)*p+1 must equal -1 and +1 modulo 3, the number 2^(k+1)*p must be 0 modulo 3, implying that p=3. Therefore every term, except 4, must be of the form 3*2^k such that 3*2^(k+1)-1, 3*2^(k+1)+1 are twin primes. In other words, k+1 belongs to the intersection of A002253 and A002235.
According to Ballinger and Keller's lists, there are no other such k up to 22*10^6. Therefore a(6) (if it exists) is greater than 3*2^(22*10^6) ~= 10^6622660. (End)
From Daniel Forgues, Nov 23 2009: (Start)
The 2 last known anti-primes seem to relate to the Fermat primes (coincidence?):
96 = 3 * 2^5 = 3 * 2^F_1 = 3 * 2^[2^(2^1) + 1] and
393216 = 3 * 2^17 = 3 * 2^F_2 = 3 * 2^[2^(2^2) + 1],
where F_k is the k-th Fermat prime. (End)

Ray Ballinger and Wilfrid Keller, List of primes k·2^n + 1 for k < 300
Wilfrid Keller, List of primes k·2^n - 1 for k < 300
Jon Perry, The Anti-divisor [Cached copy]
Jon Perry, The Anti-divisor: Even More Anti-Divisors [Cached copy]

Cf. A000215, A002253, A002235, A066272, A019434.

antid[n_] := Select[ Union[ Join[ Select[ Divisors[2n - 1], OddQ[ # ] && # != 1 & ], Select[ Divisors[2n + 1], OddQ[ # ] && # != 1 & ], 2n/Select[ Divisors[ 2n], OddQ[ # ] && # != 1 &]]], # < n & ]; Select[ Range[10^5], Length[ antid[ # ]] == 1 & ]

Edited by Max Alekseyev, Oct 13 2009

A111046 Difference between squares of twin prime pairs.

Original entry on oeis.org

16, 24, 48, 72, 120, 168, 240, 288, 408, 432, 552, 600, 720, 768, 792, 912, 960, 1080, 1128, 1248, 1392, 1680, 1728, 1848, 2088, 2280, 2400, 2472, 2568, 2640, 3240, 3288, 3312, 3432, 3528, 4080, 4128, 4200, 4248, 4368, 4608, 4920, 5112, 5160, 5208, 5280

Offset: 1

Giovanni Teofilatto, Oct 06 2005

Except for the first term 16 = 4^2, a(n) is never a square.

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

Cf. A001359, A006512, A014574, A040040, A054735, A108570, A108604.

a111046 = (* 2) . a054735  -- Reinhard Zumkeller, Feb 10 2015

ZL:=[]:for p from 1 to 1400 do if (isprime(p) and isprime(p+2)) then ZL:=[op(ZL),(((p+2)^2)-p^2)]; fi; od; print(ZL); # Zerinvary Lajos, Mar 08 2007

Select[Table[Prime[n] + 1, {n, 220}], PrimeQ[ # + 1] &] *4 (* Ray Chandler, Oct 12 2005 *)
4+4#&/@Select[Partition[Prime[Range[300]],2,1],#[[2]]-#[[1]]==2&] [[All,1]] (* Harvey P. Dale, Apr 12 2018 *)

a(n) = A006512(n)^2 - A001359(n)^2 = A108604(n) - A108570(n) = 2*A054735(n) = 4*A014574(n) = 8*A040040(n).

Edited and extended by Ray Chandler, Oct 12 2005

A260689 Table read by rows: numbers m such that (2n-m, 2n+m) is a prime pair.

Original entry on oeis.org

1, 1, 3, 5, 3, 7, 1, 5, 7, 3, 9, 3, 13, 1, 5, 11, 13, 3, 9, 17, 9, 15, 19, 5, 7, 13, 17, 19, 3, 15, 21, 9, 15, 25, 1, 7, 11, 13, 17, 23, 9, 15, 21, 27, 29, 3, 27, 5, 7, 17, 23, 25, 31, 9, 15, 21, 33, 35, 3, 21, 27, 33, 1, 5, 11, 19, 25, 29, 31, 37, 3, 15, 27

Offset: 2

Reinhard Zumkeller, Nov 17 2015

1 <= T(n,k) <= 2*n-3; T(n,2) > 3 for n > 3; all terms are odd;
A264526(n) = T(n,1);
A264527(n) = T(n,A069360(n));
T(A040040(n),1) = 1;
T(A088763(n),1) = 3.

			.   n | T(n,k)          | (2*n-T(n,k), 2*n+T(n,k))       k=1..A069360(n)
. ----+-----------------+-----------------------------------------------
.   2 | 1               | (3,5)
.   3 | 1               | (5,7)
.   4 | 3,5             | (5,11) (3,13)
.   5 | 3,7             | (7,13) (3,17)
.   6 | 1,5,7           | (11,13) (7,17) (5,19)
.   7 | 3,9             | (11,17) (5,23)
.   8 | 3,13            | (13,19) (3,29)
.   9 | 1,5,11,13       | (17,19) (13,23) (7,29) (5,31)
.  10 | 3,9,17          | (17,23) (11,29) (3,37)
.  11 | 9,15,19         | (13,31) (7,37) (3,41)
.  12 | 5,7,13,17,19    | (19,29) (17,31) (11,37) (7,41) (5,43)
.  13 | 3,15,21         | (23,29) (11,41) (5,47)
.  14 | 9,15,25         | (19,37) (13,43) (3,53)
.  15 | 1,7,11,13,17,23 | (29,31) (23,37) (19,41) (17,43) (13,47) (7,53)
.  16 | 9,15,21,27,29   | (23,41) (17,47) (11,53) (5,59) (3,61)
.  17 | 3,27            | (31,37) (7,61)
.  18 | 5,7,17,23,25,31 | (31,41) (29,43) (19,53) (13,59) (11,61) (5,67)
.  19 | 9,15,21,33,35   | (29,47) (23,53) (17,59) (5,71) (3,73)
.  20 | 3,21,27,33      | (37,43) (19,61) (13,67) (7,73) .

Reinhard Zumkeller, Rows n = 2..1000 of triangle, flattened
Index entries for sequences related to Goldbach conjecture

Cf. A069360 (row lengths), A010051, A264526, A264527.

a260689 n k = a260689_tabf !! (n-2) !! (k-1)
a260689_row n = [m | m <- [1, 3 .. 2 * n - 3],
                     a010051' (2*n + m) == 1, a010051' (2*n - m) == 1]
a260689_tabf = map a260689_row [2..]

A088769 a(n) = A087678(n)/2.

Original entry on oeis.org

7, 10, 11, 14, 16, 19, 25, 26, 31, 35, 40, 44, 46, 49, 59, 61, 70, 74, 79, 86, 91, 94, 95, 101, 110, 116, 121, 124, 130, 136, 151, 161, 170, 179, 194, 196, 205, 215, 220, 224, 226, 229, 235, 250, 256, 266, 289, 304, 305, 311, 325, 326, 334, 341, 346, 350, 355

Offset: 1

Ray Chandler, Oct 26 2003

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..4350

Cf. A087678, A040040, A088763, A088765, A088767.

[n/2: n in [5..1500] |IsPrime(n+9) and IsPrime(n-9)]; // Vincenzo Librandi, May 22 2017

Select[Range[5, 2000], PrimeQ[# + 9] && PrimeQ[# - 9] &] / 2 (* Vincenzo Librandi, May 21 2017 *)

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

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]

cp's OEIS Frontend

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

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

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A088765 a(n) = A087696(n)/2.

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A104278 Numbers m such that 2m+1 and 2m-1 are not primes.

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A066466 Numbers having just one anti-divisor.

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A111046 Difference between squares of twin prime pairs.

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A260689 Table read by rows: numbers m such that (2*n-m, 2*n+m) is a prime pair.

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

A260689 Table read by rows: numbers m such that (2n-m, 2n+m) is a prime pair.

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