A031505 -id:A031505 - cp's OEIS frontend

A046132 Larger member p+4 of cousin primes (p, p+4).

7, 11, 17, 23, 41, 47, 71, 83, 101, 107, 113, 131, 167, 197, 227, 233, 281, 311, 317, 353, 383, 401, 443, 461, 467, 491, 503, 617, 647, 677, 743, 761, 773, 827, 857, 863, 881, 887, 911, 941, 971, 1013, 1091, 1097, 1217, 1283, 1301, 1307, 1427, 1433

Offset: 1

Eric W. Weisstein

nonn

A pair of cousin primes are primes of the form p and p+4 (where p+2 may or may not be a prime). - N. J. A. Sloane, Mar 18 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Cousin Primes
Wikipedia, Cousin prime.

Essentially the same as A031505. Cf. A023200, A029710, A098429.

Cf. A000040, A010051.

a046132 n = a046132_list !! (n-1)
a046132_list = filter ((== 1) . a010051') $ map (+ 4) a000040_list
-- Reinhard Zumkeller, Aug 01 2014

lst={};Do[p=Prime[n];If[PrimeQ[p4=p+4], (*Print[p4];*)AppendTo[lst, p4]], {n, 10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)
Select[Prime[Range[300]],PrimeQ[#+4]&]+4 (* Harvey P. Dale, Dec 15 2017 *)

forprime(p=2,1e5,if(isprime(p-4),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(n) = A023200(n) + 4 = A087679(n) + 2.

a(n) = 3*A157834(n-1) + 2 = A029710(n-1) + 4 = 6*A056956(n-1) + 5 (thus a(n) mod 6 == 5), for all n>1. - M. F. Hasler, Jan 15 2013

A029710 Primes such that next prime is 4 greater.

Original entry on oeis.org

7, 13, 19, 37, 43, 67, 79, 97, 103, 109, 127, 163, 193, 223, 229, 277, 307, 313, 349, 379, 397, 439, 457, 463, 487, 499, 613, 643, 673, 739, 757, 769, 823, 853, 859, 877, 883, 907, 937, 967, 1009, 1087, 1093, 1213, 1279, 1297, 1303, 1423, 1429

Offset: 1

N. J. A. Sloane

nonn

Union with A124588 gives A124589. - Reinhard Zumkeller, Dec 23 2006
For any prime p > 3, if p + 4 is prime then necessarily it is the next prime. But there cannot be three consecutive primes with mutual distance 4: If p and p + 4 are prime, then p+8 is an odd multiple of 3 (cf. formula). - M. F. Hasler, Jan 15 2013
The smaller members p of cousin prime pairs (p,p+4) excluding p=3. - Marc Morgenegg, Apr 19 2016

			79 is a term as the next prime is 79 + 4 = 83. 3 is not a term even though 3 + 4 = 7 is prime, since it is not the next one.

Marius A. Burtea, Table of n, a(n) for n = 1..14741 ( first 1000 terms from R. Zumkeller )
Index entries for primes, gaps between

Essentially the same as A023200.

Cf. A001359, A029708, A031505, A124588, A124589, A157834.

p=primes(1700);m=1;
for u=1:length(p)-4
   if and(isprime(p(u)+4)==1,p(u+1)==p(u)+4);sol(m)=p(u);m=m+1;end
end
sol % Marius A. Burtea, Jan 24 2019

[p:p in PrimesUpTo(1700)| IsPrime(p+4) and NextPrime(p) eq p+4] // Marius A. Burtea, Jan 24 2019

for i from 1 to 226 do if ithprime(i+1) = ithprime(i) + 4 then print({ithprime(i)}); fi; od; # Zerinvary Lajos, Mar 19 2007

Select[Prime[Range[225]], NextPrime[#] == # + 4 &] (* Alonso del Arte, Jan 17 2013 *)
Transpose[Select[Partition[Prime[Range[300]],2,1],#[[2]]-#[[1]]==4&]] [[1]] (* Harvey P. Dale, Mar 28 2016 *)

forprime(p=1, 1e4, if(nextprime(p+1)-p==4, print1(p, ", "))) \\ Felix Fröhlich, Aug 16 2014

a(n) = A031505(n + 1) - 4 = A029708(n) - 2.

a(n) = 1 (mod 6) for all n; (a(n) + 2)/3 = A157834(n), i.e., a(n) = 3*A157834(n) - 2. - M. F. Hasler, Jan 15 2013

A029708 Numbers k such that k-2 and k+2 are consecutive primes.

Original entry on oeis.org

9, 15, 21, 39, 45, 69, 81, 99, 105, 111, 129, 165, 195, 225, 231, 279, 309, 315, 351, 381, 399, 441, 459, 465, 489, 501, 615, 645, 675, 741, 759, 771, 825, 855, 861, 879, 885, 909, 939, 969, 1011, 1089, 1095, 1215, 1281, 1299, 1305, 1425, 1431

Offset: 1

N. J. A. Sloane

nonn

All terms are multiples of 3. Minimal first difference is 6. - Zak Seidov, May 15 2013

Marius A. Burtea, Table of n, a(n) for n = 1..10548 (first 1000 terms from Zak Seidov )
Zak Seidov, Table of n, a(n)/3 for n = 1..1000

Essentially the same as A087679.

[k:k in [1..1500]| IsPrime(k-2) and NextPrime(k-2) eq k+2 ]; // Marius A. Burtea, Jan 24 2019

f[n_]:=PrimeQ[n-2]&&PrimeQ[n+2]; lst={};Do[If[f[n],AppendTo[lst,n]],{n,7,8!,2}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 09 2009 *)
Select[Range[9,1432,6],PrimeQ[#-2]&&PrimeQ[#+2]&] (* Zak Seidov, May 15 2013 - just for terms in DATA *)
Mean/@Select[Partition[Prime[Range[300]],2,1],#[[2]]-#[[1]]==4&] (* Harvey P. Dale, Feb 15 2020 *)

a(n) = (A029710(n) + A031505(n+1))/2 = A029710(n) + 2 = A031505(n+1) - 2.

A067830 Primes p such that sigma(p-4) < p.

Original entry on oeis.org

5, 7, 11, 17, 23, 41, 47, 71, 83, 101, 107, 113, 131, 167, 197, 227, 233, 281, 311, 317, 353, 383, 401, 443, 461, 467, 491, 503, 617, 647, 677, 743, 761, 773, 827, 857, 863, 881, 887, 911, 941, 971, 1013, 1091, 1097, 1217, 1283, 1301, 1307, 1427, 1433, 1451

Offset: 1

Benoit Cloitre, Feb 08 2002

Except for the first term, terms are primes of the form p+4 with p prime, i.e., the sequence is essentially A031505, A046132. In other words, the solutions to sigma(x) < x + 4 are 1,2,4 and the odd primes. - Ralf Stephan, Feb 09 2004

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

Cf. A000203, A025584, A046022, A067833.

Select[Prime[Range[3, 230]], DivisorSigma[1, #-4] < # &] (* Amiram Eldar, Apr 25 2025 *)

isok(p) = isprime(p) && (p>4) && (sigma(p-4) < p); \\ Michel Marcus, Feb 15 2021

Edited by Charles R Greathouse IV, Mar 19 2010

A118590 Larger of two consecutive primes whose positive difference is a square.

Original entry on oeis.org

3, 11, 17, 23, 41, 47, 71, 83, 101, 107, 113, 131, 167, 197, 227, 233, 281, 311, 317, 353, 383, 401, 443, 461, 467, 491, 503, 617, 647, 677, 743, 761, 773, 827, 857, 863, 881, 887, 911, 941, 971, 1013, 1091, 1097, 1217, 1283, 1301, 1307, 1427, 1433, 1451, 1487

Offset: 1

Cino Hilliard, May 07 2006

			7 and 11 are consecutive primes. 11-7 = 4 a square, so 11 is the second term in the table.

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for primes, gaps between

Cf. A031935, A031505, A134117 (gap 6^2), A204670 (gap 8^2), A050434 (gap 10^2), A138198, A161002.

Select[Table[Prime[n], {n, 2, 237}], IntegerQ[Sqrt[# - Prime[PrimePi[# - 1]]]] &] (* Jayanta Basu, Apr 23 2013 *)
nn = 500; ps = Prime[Range[nn]]; t = {}; Do[If[IntegerQ[Sqrt[ps[[n]] - ps[[n-1]]]], AppendTo[t, ps[[n]]]], {n, 2, nn}]; t (* T. D. Noe, Apr 23 2013 *)
Prime[#+1]&/@Flatten[Position[Differences[Prime[Range[250]]],?(IntegerQ[ Sqrt[#]]&)]] (* _Harvey P. Dale, May 08 2019 *)

g(n) = for(x=2, n, if(issquare(prime(x)-prime(x-1)), print1(prime(x)",")))

Superset of A031935 and A031505. [From R. J. Mathar, Aug 08 2008]

A111981 Numbers n such that 2n-1 and 2n+3 are consecutive primes.

Original entry on oeis.org

4, 7, 10, 19, 22, 34, 40, 49, 52, 55, 64, 82, 97, 112, 115, 139, 154, 157, 175, 190, 199, 220, 229, 232, 244, 250, 307, 322, 337, 370, 379, 385, 412, 427, 430, 439, 442, 454, 469, 484, 505, 544, 547, 607, 640, 649, 652, 712, 715, 724, 742, 745, 775, 784, 790

Offset: 1

Ray Chandler, Aug 24 2005

nonn

Essentially the same as A088762.

a(n) = (A029708(n)-1)/2 = (A029710(n)+1)/2 = (A031505(n)-3)/2.

A111980 Union of pairs of consecutive primes p, q with q-p = 4.

Original entry on oeis.org

7, 11, 13, 17, 19, 23, 37, 41, 43, 47, 67, 71, 79, 83, 97, 101, 103, 107, 109, 113, 127, 131, 163, 167, 193, 197, 223, 227, 229, 233, 277, 281, 307, 311, 313, 317, 349, 353, 379, 383, 397, 401, 439, 443, 457, 461, 463, 467, 487, 491, 499, 503, 613, 617, 643

Offset: 1

Ray Chandler, Aug 24 2005

nonn

Essentially the same as A094343.
Union of A029710 and A031505.
Cf. A140546.

Flatten[Select[Partition[Prime[Range[200]],2,1],#[[2]]-#[[1]]==4&]]//Union (* Harvey P. Dale, Jul 09 2024 *)

A126720 Primes p such that p - q = 24, where q is the previous prime before p; or prime numbers preceded by precisely 23 composite numbers.

Original entry on oeis.org

1693, 2203, 4201, 4547, 4783, 5261, 6197, 6421, 6761, 7103, 7393, 7817, 8147, 8353, 9091, 11027, 11657, 11863, 12097, 12143, 13033, 13291, 16057, 16217, 16477, 16787, 16811, 17077, 17707, 18013, 18617, 18661, 19207, 19531, 20507, 22433, 22901

Offset: 1

Artur Jasinski, Feb 13 2007

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

Cf. A000230, A033560, A031505, A031925, A031927, A098974, A098976, A061779.

a = {}; Do[If[Prime[x + 1] - Prime[x] == 24, AppendTo[a, Prime[x + 1]]], {x, 1, 10000}]; a

q=2; forprime(p=3,1e5, if(p-q==24, print1(p", ")); q=p) \\ Charles R Greathouse IV, Mar 13 2020

a(n) = A098974(n) + 24. - Amiram Eldar, Mar 13 2020

a(n) >> n log^2 n. - Charles R Greathouse IV, Mar 13 2020

cp's OEIS Frontend

A046132 Larger member p+4 of cousin primes (p, p+4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A029710 Primes such that next prime is 4 greater.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

MATLAB

Magma

Maple

Mathematica

PARI

Formula

A029708 Numbers k such that k-2 and k+2 are consecutive primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A067830 Primes p such that sigma(p-4) < p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A118590 Larger of two consecutive primes whose positive difference is a square.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A111981 Numbers n such that 2n-1 and 2n+3 are consecutive primes.

Views

Author

Keywords

Crossrefs

Formula

A111980 Union of pairs of consecutive primes p, q with q-p = 4.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A126720 Primes p such that p - q = 24, where q is the previous prime before p; or prime numbers preceded by precisely 23 composite numbers.

Views

Author

Keywords

Links

Crossrefs