A157834 -id:A157834 - 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

A087679 Numbers k such that both k+2 and k-2 are prime.

Original entry on oeis.org

5, 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, 1449, 1485

Offset: 1

Zak Seidov, Sep 27 2003

Essentially the same as A029708: a(n) = A029708(n-1) for n>=2.
Midpoint of cousin prime pairs.
The only prime is 5. All other terms are multiples of 3. - Zak Seidov, May 19 2014

M. F. Hasler, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Cousin Primes

Cf. A014574, A087678-A087683, A087695-A087697, A088762.

ZL:=[]:for p from 1 to 1485 do if (isprime(p) and isprime(p+4) ) then ZL:=[op(ZL),(p+(p+4))/2]; fi; od; print(ZL); # Zerinvary Lajos, Mar 07 2007

lst={};Do[If[PrimeQ[n-2]&&PrimeQ[n+2],AppendTo[lst,n]],{n,3,8!,2}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 14 2009 *)

s=[]; for(n=1, 2000, if(isprime(n-2) && isprime(n+2), s=concat(s, n))); s \\ Colin Barker, May 19 2014

is_A087679(n)={isprime(n-2) && isprime(n+2)} \\ For numbers >> 10^12 one should add conditions {n%6==3 && ... || n==5} or consider only such numbers congruent to 3 (mod 6). - M. F. Hasler, Apr 05 2017

a(n) = (A023200(n) + A046132(n))/2 = A023200(n) + 2 = A046132(n) - 2.

a(n+1) = A056956(n)*6 + 3 = A157834(n)*3; a(n) = A088762(n)*2 + 1. - M. F. Hasler, Apr 05 2017

More terms from Ray Chandler, Oct 26 2003

A268475 Numbers n such that n^3 +/- 2 and 3*n +/- 2 are all prime.

Original entry on oeis.org

435, 555, 2415, 31635, 38025, 44835, 80625, 88335, 97455, 98505, 99435, 124335, 142065, 145095, 165375, 176055, 204765, 246435, 279225, 293475, 310095, 315555, 332085, 344745, 348735, 376935, 392415, 443595, 462105, 467385, 482355, 581415, 609555, 626775, 636015

Offset: 1

K. D. Bajpai, Feb 05 2016

nonn

All the terms in this sequence are congruent to 0 (mod 5).
Each term in this sequence yields two sets of cousin prime pairs i.e., for n = 435 -> {82312877, 82312873} and {1307, 1303}.
All terms are congruent to 15 mod 30. - Robert Israel, Feb 05 2016

			435 is in the sequence because 435^3 + - 2 =  82312877, 82312873; 3*435 + - 2 = 1307, 1303 are all prime.
555 is in the sequence because 555^3 + - 2 =  170953877, 170953873; 3*555 + - 2 = 1667, 1663 are all prime.

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A024893, A090121, A108701, A153183, A157834.

[n : n in [1..1e5] | IsPrime(n^3 + 2) and IsPrime(n^3 - 2) and IsPrime(3*n + 2) and IsPrime(3*n - 2)];

select(n -> andmap(isprime, [n^3 + 2, n^3 - 2, 3*n + 2, 3*n - 2]), [seq(p, p=1.. 10^6)]);

Select[Range[1000000], PrimeQ[#^3 + 2] && PrimeQ[#^3 - 2] && PrimeQ[3 # + 2] && PrimeQ[3 # - 2] &]

for(n = 1,1e5, if( isprime(n^3 + 2) && isprime(n^3 - 2) && isprime(3*n + 2) && isprime(3*n - 2), print1(n ", ")))

A233348 Numbers n such that 3n+2 and 3n-2 are both prime for n multiple of 5 (A008587).

Original entry on oeis.org

5, 15, 35, 55, 65, 75, 105, 155, 205, 215, 225, 275, 285, 295, 365, 405, 435, 475, 495, 555, 565, 595, 625, 665, 695, 735, 765, 825, 895, 945, 985, 1055, 1085, 1115, 1155, 1205, 1225, 1265, 1315, 1335, 1385, 1505, 1595, 1605, 1645, 1745, 1805, 1835, 1885

Offset: 1

César Aguilera, Dec 07 2013

			For n=15; 3*15+2=47 and 3*15-2=43.

Cf. A157834 (n such that 3n-2 and 3n+2 are both prime).

Select[5*Range[500], PrimeQ[3 # + 2] && PrimeQ[3 # - 2] &] (* T. D. Noe, Dec 09 2013 *)

Intersection of A008587 and A157834.

cp's OEIS Frontend

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

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A029710 Primes such that next prime is 4 greater.

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A087679 Numbers k such that both k+2 and k-2 are prime.

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A268475 Numbers n such that n^3 +/- 2 and 3*n +/- 2 are all prime.

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Magma

Maple

Mathematica

PARI

A233348 Numbers n such that 3*n+2 and 3*n-2 are both prime for n multiple of 5 (A008587).

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Examples

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Mathematica

Formula

A233348 Numbers n such that 3n+2 and 3n-2 are both prime for n multiple of 5 (A008587).