A124589 -id:A124589 - cp's OEIS frontend

A029710 Primes such that next prime is 4 greater.

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

A124582 Primes p such that q-p >= 6, where q is the next prime after p.

Original entry on oeis.org

23, 31, 47, 53, 61, 73, 83, 89, 113, 131, 139, 151, 157, 167, 173, 181, 199, 211, 233, 241, 251, 257, 263, 271, 283, 293, 317, 331, 337, 353, 359, 367, 373, 383, 389, 401, 409, 421, 433, 443, 449, 467, 479, 491, 503, 509, 523, 541, 547, 557, 563, 571, 577

Offset: 1

N. J. A. Sloane, Dec 19 2006

nonn

R. Zumkeller, Table of n, a(n) for n = 1..1000
K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc., 44 (2007), 1-18.
Index entries for primes, gaps between

Complement of A124589.

Cf. A134099, A134100, A134101.

d:=6; M:=1000; t0:=[]; for n from 1 to M do p:=ithprime(n); if nextprime(p) - p >= d then t0:=[op(t0),p]; fi; od: t0;

t={};q=6;Do[If[Prime[n+6]-Prime[n+5]>=q,AppendTo[t,Prime[n+5]]],{n,1,200}];t (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)
Transpose[Select[Partition[Prime[Range[200]],2,1],#[[2]]-#[[1]] >= 6&]] [[1]] (* Harvey P. Dale, May 15 2013 *)

is(n)=!isprime(n+2) && !isprime(n+4) && n>2 && isprime(n) \\ Charles R Greathouse IV, Jun 04 2015

a(n) ~ n log n. - Charles R Greathouse IV, Jun 04 2015

A124588 Primes p such that q - p <= 2, where q is the next prime after p.

Original entry on oeis.org

2, 3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607

Offset: 1

N. J. A. Sloane, Dec 19 2006; edited May 15 2008 at the suggestion of R. J. Mathar

nonn

Consists of 2 together with the lower members of twin primes, A001359. See the latter entry for references.
"Assuming certain (admittedly difficult) conjectures on the distribution of primes in arithmetic progressions, [Goldston-Pintz-Yildirim] prove the existence of infinitely many prime pairs that differ at most by 16." - Soundararajan
Lesser of twin primes together with 2; union with A029710 gives A124589. - Reinhard Zumkeller, Dec 23 2006
Primes p such that either p + 3/2 +- 1/2 is prime. - Juri-Stepan Gerasimov, Jan 29 2010
The prime differences of 2 primes (without repetition). - Juri-Stepan Gerasimov, Jun 01 2010, Jun 08 2010
Numbers k such that sigma(k*(k+2)) = (k+1)*(k+3). - Wesley Ivan Hurt, May 08 2022

K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc., 44 (2007), 1-18.
Index entries for primes, gaps between

Cf. A001359.

Transpose[Select[Partition[Prime[Range[300]],2,1],#[[2]]-#[[1]]<3&]] [[1]] (* Harvey P. Dale, Feb 11 2015 *)

twinl(n) = { c=0; x=1; while(cCino Hilliard, Mar 29 2008

A124590 Primes p such that q-p <= 6, where q is the next prime after p.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 97, 101, 103, 107, 109, 127, 131, 137, 149, 151, 157, 163, 167, 173, 179, 191, 193, 197, 223, 227, 229, 233, 239, 251, 257, 263, 269, 271, 277, 281, 307, 311, 313, 331, 347, 349, 353, 367

Offset: 1

N. J. A. Sloane, Dec 19 2006

Goldston, Graham, Pintz, & Yilidirm give a conditional proof that this sequence is infinite; see their Theorem 4. - Charles R Greathouse IV, Jul 31 2013

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
D. A. Goldston, S. W. Graham, J. Pintz, C. Y. Yilidirm, Small gaps between primes or almost primes (2005)
K. Soundararajan, Small gaps between prime numbers: the work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc. 44 (2007), pp. 1-18.
Index entries for primes, gaps between

Cf. A083371, A124589, A031924.

v=List([2]);p=3;forprime(q=5,1e3,if(q-p<=6,listput(v,p));p=q);Vec(v) \\ Charles R Greathouse IV, Jul 31 2013

list(lim)=my(v=List(),p=2); forprime(q=3,nextprime(lim\1+1), if(q-p<7, listput(v,p)); p=q); Vec(v) \\ Charles R Greathouse IV, Jan 31 2017

A000040 MINUS A083371. - R. J. Mathar, Jun 15 2008
A124589 UNION A031924. - R. J. Mathar, Jan 23 2022
a(n) >> n log^2 n. - Charles R Greathouse IV, Jan 31 2017

cp's OEIS Frontend

A029710 Primes such that next prime is 4 greater.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

MATLAB

Magma

Maple

Mathematica

PARI

Formula

A124582 Primes p such that q-p >= 6, where q is the next prime after p.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A124588 Primes p such that q - p <= 2, where q is the next prime after p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A124590 Primes p such that q-p <= 6, where q is the next prime after p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula