A029710 -id:A029710 - cp's OEIS frontend

A098974 Primes p such that q-p = 24, where q is the next prime after p.

1669, 2179, 4177, 4523, 4759, 5237, 6173, 6397, 6737, 7079, 7369, 7793, 8123, 8329, 9067, 11003, 11633, 11839, 12073, 12119, 13009, 13267, 16033, 16193, 16453, 16763, 16787, 17053, 17683, 17989, 18593, 18637, 19183, 19507, 20483, 22409, 22877, 23227

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Oct 23 2004

Lower prime of a difference of 24 between consecutive primes.

23 successive numbers after prime number p are composite. - Artur Jasinski, Jan 15 2007

Remi Eismann, Table of n, a(n) for n = 1..10000
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. A000040, A001223, A054541, A075526, A001359, A054799, A063091, A096292, A015913, A023200, A029710, A031934, A031936, A031938.

Cf. A000230, A023200, A031924, A031926, A031928, A031930, A031932, A061779.

a = {}; Do[If[Prime[x + 1] - Prime[x] == 24, AppendTo[a, Prime[x]]], {x, 1, 10000}]; a (* Artur Jasinski, Jan 15 2007 *)

Entry revised by N. J. A. Sloane, Feb 13 2007

A126784 Primes p such that q-p = 32, where q is the next prime after p.

Original entry on oeis.org

5591, 10799, 27701, 27851, 33647, 39047, 41081, 41687, 43721, 44417, 45989, 47459, 50789, 52457, 55259, 55547, 61781, 62351, 64817, 66239, 67307, 69959, 73907, 79907, 80567, 82307, 84089, 88037, 94169, 94961, 99191, 99929, 100559, 102611

Offset: 1

Douglas Winston (douglas.winston(AT)srupc.com), Feb 24 2007

Lower prime of a difference of 32 between consecutive primes.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001359, A029710, A031924, A031926, A031928, A031930, A031932, A031934, A031936, A031938, A061779, A098974, A124594, A124595, A124596, A000230, A050434.

lista(nn) = {p = 2; while (p < nn, q = nextprime(p+1); if (q - p == 32, print1(p, ", ")); p = q;);} \\ Michel Marcus, Jul 17 2013

A274121 The gap prime(n+1) - prime(n) occurs for the a(n)-th time.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 3, 1, 5, 2, 4, 6, 5, 3, 4, 7, 5, 6, 8, 6, 7, 7, 1, 8, 9, 9, 10, 10, 1, 11, 8, 11, 1, 12, 9, 10, 12, 11, 12, 13, 2, 14, 13, 15, 1, 2, 14, 16, 15, 13, 17, 3, 14, 15, 16, 18, 17, 16, 19, 4, 2, 17, 20, 18, 3, 18, 5, 21, 19, 19, 2, 20, 21, 20, 22, 3, 21, 4, 6, 22, 7, 23, 23, 22, 24, 5, 23, 24, 24, 3, 6

Offset: 1

David A. Corneth, Jun 10 2016

nonn

Terms of this sequence grow without bound; any even number occurs in this sequence. Zhang proved that there are infinitely many primes 4680 apart from each other (see link "Bounded gaps between primes").
For a conjectured count of gap n below x, see link Polignac's conjecture.
Polignac's conjecture states that "For any positive even number n, there are infinitely many prime gaps of size n.". By this conjecture, every positive apppears infinitely many times in this sequence (see link "Polignac's conjecture").

			(p, g) denotes a prime p and the gap up to the next prime. So p + g is the next prime after p. These pairs start (2, 1), (3, 2), (5, 2), (7, 4), (11, 2). From here we see that:
- the gap after the first prime, 1 occurs for the first time, so a(1) = 1.
- the gap after the second prime, 2, occurs for the first time, so a(2) = 1.
- the gap after the third prime, 2, occurs for the second time, so a(3) = 2.
- the gap after the fourth prime, 4, occurs for the first time, so a(4) = 1.
- the gap after the fifth prime, 2, occurs for the third time, so a(5) = 3.

David A. Corneth, Table of n, a(n) for n = 1..100021 (all terms up to 1300000)
David A. Corneth, PARI program
PolyMath, Bounded gaps between primes.
Wikipedia, Polignac's conjecture.

Cf. A000230, A005597, A001359, A029710, A274122, A274123.

\\ See link by name "PARI program" for an extended version with comments.
upto(n) = {my(gapcount=List(), freqgap = List([1])); n = max(n, 3); forprime(i=3,n,
g = nextprime(i+1) - i; for(i=#gapcount+1, g\2, listput(gapcount,0));  gapcount[g\2]++; listput(freqgap,gapcount[g\2]));freqgap} \\ David A. Corneth, Jun 28 2016

a(primepi(A000230(n))) = 1.
a(primepi(A001359(n))) = n.
a(primepi(A029710(n))) = n.

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

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.

A157834 Numbers n such that 3n-2 and 3n+2 are both prime.

Original entry on oeis.org

3, 5, 7, 13, 15, 23, 27, 33, 35, 37, 43, 55, 65, 75, 77, 93, 103, 105, 117, 127, 133, 147, 153, 155, 163, 167, 205, 215, 225, 247, 253, 257, 275, 285, 287, 293, 295, 303, 313, 323, 337, 363, 365, 405, 427, 433, 435, 475, 477, 483, 495, 497, 517

Offset: 1

Kyle D. Balliet, Mar 07 2009

nonn

Barycenter of cousin primes (A029708; see also A029710, A023200, A046132), divided by 3. When p>3 and p+4 both are prime, then p = 1 (mod 6) and p+2 = 3 (mod 6). - M. F. Hasler, Jan 14 2013

			15*3 +/- 2 = 43,47 (both prime).

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A024893 and A153183.

[n: n in [1..1000]|IsPrime(3*n-2)and IsPrime(3*n+2)] // Vincenzo Librandi, Dec 13 2010

select(t -> isprime(3*t+2) and isprime(3*t-2), [seq(t,t=3..1000,2)]); # Robert Israel, May 28 2017

Select[Range[600],AllTrue[3#+{2,-2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 03 2019 *)

Intersection of A024893 and A153183.
a(n) = A029708(n)/3. - Zak Seidov, Aug 07 2009
a(n) = A056956(n)*2+1 = (A029710(n)+2)/3 = (A023200(n+1)+2)/3. - M. F. Hasler, Jan 14 2013

A346239 Möbius transform of A341512, sigma(n)A003961(n) - nsigma(A003961(n)).

Original entry on oeis.org

0, 1, 2, 10, 2, 33, 4, 74, 44, 55, 2, 278, 4, 115, 116, 490, 2, 613, 4, 498, 242, 169, 6, 1942, 92, 265, 742, 1046, 2, 1591, 6, 3086, 344, 355, 330, 4986, 4, 487, 542, 3570, 2, 3347, 4, 1638, 2326, 737, 6, 12542, 376, 2121, 716, 2546, 6, 9869, 388, 7510, 986, 943, 2, 12894, 6, 1225, 4872, 18970, 630, 5353, 4, 3498, 1492

Offset: 1

Antti Karttunen, Jul 13 2021

nonn

Cf. A000203, A003961, A008683, A341528, A341529, A341512, A346240, A347096, A347124, A347125.

Cf. also the sequences A001359, A029710, A031924 that give the positions of 2's, 4's and 6's in this sequence, or at least subsets of such positions.

A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341528(n) = (n*sigma(A003961(n)));
A341529(n) = (sigma(n)*A003961(n));
A341512(n) = (A341529(n)-A341528(n));
A346239(n) = sumdiv(n,d,moebius(n/d)*A341512(d));

a(n) = Sum_{d|n} A008683(n/d) * A341512(d).
a(n) = A341512(n) - A346240(n).
a(n) = A347125(n) - A347124(n). - Antti Karttunen, Aug 25 2021

A078561 p, p+4 and p+10 are consecutive primes.

Original entry on oeis.org

19, 43, 79, 127, 163, 229, 349, 379, 439, 499, 643, 673, 937, 967, 1009, 1093, 1213, 1279, 1429, 1489, 1549, 1597, 1609, 2203, 2347, 2389, 2437, 2539, 2689, 2833, 2953, 3079, 3319, 3529, 3613, 3793, 3907, 3919, 4003, 4129, 4447, 4639, 4789, 4933, 4999

Offset: 1

Labos Elemer, Dec 10 2002

nonn

Subsequence of A029710. - R. J. Mathar, May 06 2017

			Between p and p+10 [46] difference-pattern: 19(4)23(6)29;

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. analogous inter-prime d-patterns with d<=6: A022004[24], A022005[42], A049437[26], A049438[62], A078561[46], A078562[64], A047948[66].

Select[Prime@ Range[10^3], Differences@ NestList[NextPrime, #, 2] == {4, 6} &] (* Michael De Vlieger, May 06 2017 *)
Select[Partition[Prime[Range[700]],3,1],Differences[#]=={4,6}&][[All,1]] (* Harvey P. Dale, Mar 24 2018 *)

isok(p) = isprime(p) && (nextprime(p+1) == p+4) && (nextprime(p+5) == p+10); \\ Michel Marcus, Dec 20 2013

is(n)=isprime(n) && isprime(n+4) && isprime(n+10) && !isprime(n+6) && n>3 \\ Charles R Greathouse IV, Dec 20 2013

A124589 Primes p such that q-p <= 4, where q is the next prime after p.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 29, 37, 41, 43, 59, 67, 71, 79, 97, 101, 103, 107, 109, 127, 137, 149, 163, 179, 191, 193, 197, 223, 227, 229, 239, 269, 277, 281, 307, 311, 313, 347, 349, 379, 397, 419, 431, 439, 457, 461, 463, 487, 499, 521, 569, 599, 613, 617, 641, 643, 659

Offset: 1

N. J. A. Sloane, Dec 19 2006

nonn

Union of A124588 and A029710; complement of A124582. - Reinhard Zumkeller, Dec 23 2006

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

CF. A029710, A124582, A124588.

Transpose[Select[Partition[Prime[Range[200]],2,1],Last[#]-First[#]<5&]][[1]] (* Harvey P. Dale, Apr 22 2013 *)

is(n)=isprime(n) && (isprime(n+2) || isprime(n+4) || n==2) \\ Charles R Greathouse IV, Jun 01 2016

a(n) >> n log^2 n. Infinite under standard conjectures. - Charles R Greathouse IV, Jun 01 2016

A076976 Product of the smallest prime divisors of composite numbers between successive primes.

Original entry on oeis.org

1, 2, 2, 12, 2, 12, 2, 12, 120, 2, 120, 12, 2, 12, 168, 120, 2, 120, 12, 2, 168, 12, 120, 1680, 12, 2, 12, 2, 12, 2217600, 12, 168, 2, 15840, 2, 120, 168, 12, 312, 120, 2, 15840, 2, 12, 2, 221760, 262080, 12, 2, 12, 120, 2, 18720, 264, 168, 120, 2, 120, 12, 2, 34272

Offset: 1

Amarnath Murthy, Oct 23 2002

nonn

From Bernard Schott, Apr 09 2020: (Start)
a(n) = 2 iff prime(n) is in A001359 (prime gap=2).
a(n) = 12 iff prime(n) is in A029710 (prime gap=4).
a(n) = 24 * p with p prime >= 5 iff prime(n) is in A031924 (prime gap=6).
a(n) = 2^m * q with q odd >= 3 iff prime(n+1) - prime(n) = 2*m where m = A007814(a(n)). (End)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A029707 (a(n)=2), A029709 (a(n)=12), A076977.

Cf. A001359, A029710, A031924.

p:= 2:
for i from 1 to 100 do
  q:= p; p:= nextprime(p);
  A[i]:= mul(min(numtheory:-factorset(i)),i=q+1..p-1);
od:
seq(A[i],i=1..100); # Robert Israel, Mar 30 2020

pspd[{p1_,p2_}]:=Times@@(FactorInteger[#][[1,1]]&/@Range[p1+1,p2-1]); pspd/@Partition[ Prime[Range[70]],2,1] (* Harvey P. Dale, Jan 12 2024 *)

a(n) = {my(p=1, pn=prime(n)); forcomposite(c=pn, nextprime(pn+1)-1, p *= vecmin(factor(c)[,1]);); p;} \\ Michel Marcus, Mar 31 2020

More terms from Sascha Kurz, Jan 22 2003

cp's OEIS Frontend

A098974 Primes p such that q-p = 24, where q is the next prime after p.

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A126784 Primes p such that q-p = 32, where q is the next prime after p.

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PARI

A274121 The gap prime(n+1) - prime(n) occurs for the a(n)-th time.

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

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PARI

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

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Magma

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A157834 Numbers n such that 3n-2 and 3n+2 are both prime.

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Magma

Maple

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A346239 Möbius transform of A341512, sigma(n)*A003961(n) - n*sigma(A003961(n)).

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A078561 p, p+4 and p+10 are consecutive primes.

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A346239 Möbius transform of A341512, sigma(n)A003961(n) - nsigma(A003961(n)).