A023200 -id:A023200 - cp's OEIS frontend

A049489 Primes p such that p + 32 is also prime.

5, 11, 29, 41, 47, 71, 107, 131, 149, 167, 179, 191, 197, 239, 251, 281, 317, 347, 389, 401, 431, 467, 491, 509, 569, 587, 599, 641, 659, 677, 701, 719, 797, 821, 827, 887, 977, 1019, 1031, 1061, 1091, 1097, 1181, 1217, 1259, 1289, 1367, 1427, 1439, 1451

Offset: 1

Labos Elemer

			29 and 29 + 32 = 61 are both prime.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000

Cf. A001359, A023200, A023202, A049488.

[p: p in PrimesUpTo(2000) | IsPrime(p+32)]; // Vincenzo Librandi, Apr 22 2015

Primes:= select(isprime,{seq(i,i=3..10000,2)}):
sort(convert(Primes intersect map(`-`,Primes,32),list)); # Robert Israel, Dec 20 2015

Select[Range[2000], PrimeQ[#] && PrimeQ[# + 32] &] (* Vincenzo Librandi, Apr 22 2015 *)
Select[Prime[Range[300]],PrimeQ[#+32]&] (* Harvey P. Dale, Oct 14 2017 *)

isok(n) = isprime(n) && isprime(n+32); \\ Michel Marcus, Dec 31 2013

Name improved by Bruno Berselli, Apr 22 2015

A049490 a(n) and a(n)+64 both prime.

Original entry on oeis.org

3, 7, 19, 37, 43, 67, 73, 103, 109, 127, 163, 193, 199, 229, 283, 337, 367, 379, 397, 439, 457, 499, 523, 577, 613, 619, 709, 733, 757, 823, 877, 883, 907, 919, 967, 997, 1033, 1039, 1087, 1117, 1123, 1129, 1153, 1213, 1237, 1297, 1303, 1423, 1429, 1447

Offset: 1

Labos Elemer

nonn

			19 and 19+64=83 both prime.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A001359, A023200, A023202.

Select[Prime[Range[300]],PrimeQ[#+64]&] (* Harvey P. Dale, Mar 21 2011 *)

select(p -> isprime(p+64),primes(1000)) \\ Edward Jiang, Sep 05 2014

A094343 List of pairs of primes (p, q) with q - p = 4.

Original entry on oeis.org

3, 7, 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

Gerard Schildberger, Jun 04 2004

The two primes p and p+4 are not necessarily consecutive primes (for that, see A111980).

The pairs are listed in order, sorted by their smallest member. - N. J. A. Sloane, Dec 27 2019

			The pairs are (3,7), (7,11), (13,17), etc.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Cousin Primes

Almost identical to A111980.
Union of A023200 and A046132.
Cf. twin primes (A001097).
See also A000040, A111981, A001097.
For a gap of 6 (which initially is very common) see A140546.

Flatten[{#,#+4}&/@Select[Prime[Range[200]],PrimeQ[#+4]&]] (* Harvey P. Dale, Apr 13 2011 *)

isok(n) = (isprime(n) && isprime(n+4)) || (isprime(n-4) && isprime(n)); \\ Michel Marcus, Aug 26 2013

a(2*n-1)=A023200(n). a(2*n)=A046132(n).

Description was corrupted up during editing; correct description restored Aug 21 2005.

a(3) = 7 added by Vincenzo Librandi, May 06 2016

A156104 Primes p such that p+36 is also prime.

Original entry on oeis.org

5, 7, 11, 17, 23, 31, 37, 43, 47, 53, 61, 67, 71, 73, 101, 103, 113, 127, 131, 137, 157, 163, 191, 193, 197, 227, 233, 241, 257, 271, 277, 281, 311, 313, 317, 331, 337, 347, 353, 373, 383, 397, 421, 431, 443, 463, 467, 487, 521, 541, 557, 563, 571, 577, 607

Offset: 1

Vincenzo Librandi, Feb 08 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A156112.

Cf. sequences of the type p+n are primes: A001359 (n=2), A023200 (n=4), A023201 (n=6), A023202 (n=8), A023203 (n=10), A046133 (n=12), A153417 (n=14), A049488 (n=16), A153418 (n=18), A153419 (n=20), A242476 (n=22), A033560 (n=24), A252089 (n=26), A252090 (n=28), A049481 (n=30), A049489 (n=32), A252091 (n=34), this sequence (n=36); A062284 (n=50), A049490 (n=64), A156105 (n=72), A156107 (n=144).

[p: p in PrimesUpTo(1000) | IsPrime(p + 36)]; // Vincenzo Librandi, Oct 31 2012

Select[Prime[Range[1000]], PrimeQ[(#+ 36)]&] (* Vincenzo Librandi, Oct 31 2012 *)

A218867 Number of prime pairs {p,q} with p>q and {p-4,q+4} also prime such that p+(1+(n mod 6))q=n if n is not congruent to 4 (mod 6), and p-q=n and q

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 2, 1, 2, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 0, 1, 2, 2, 2, 2, 0, 2, 2, 1, 1, 1, 2, 1, 0, 0, 1, 0, 2, 2, 0, 2, 1, 3, 0, 1, 1, 2, 2, 1, 0, 3, 2, 3, 0, 2, 1, 4, 1, 1, 2, 1, 3, 2

Offset: 1

Zhi-Wei Sun, Nov 13 2012

nonn

Conjecture: a(n)>0 for all n>50000 with n different from 50627, 61127, 66503.

This conjecture implies that there are infinitely many cousin prime pairs. It is similar to the conjectures related to A219157 and A219055.

			a(20)=1 since 20=11+3*3 with 11-4 and 3+4 prime. a(28)=1 since 28=41-13 with 41-4 and 13+4 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.
Zhi-Wei Sun, Table of n, a(n) for n = 1..10^5.

Cf. A023200, A046132, A219157, A219055, A002375, A046927, A218754, A218825, A219052.

c[n_]:=c[n]=If[Mod[n+2,6]==0,1,-1-Mod[n,6]]; d[n_]:=d[n]=2+If[Mod[n+2,6]>0,Mod[n,6],0]; a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+4] == True && PrimeQ[n+c[n]Prime[k]] == True && PrimeQ[n+c[n]Prime[k]-4]==True,1,0], {k,1,PrimePi[(n-1)/d[n]]}]; Do[Print[n," ",a[n]], {n,100}]

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

Original entry on oeis.org

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

A143206 Product of the n-th cousin prime pair.

Original entry on oeis.org

21, 77, 221, 437, 1517, 2021, 4757, 6557, 9797, 11021, 12317, 16637, 27221, 38021, 50621, 53357, 77837, 95477, 99221, 123197, 145157, 159197, 194477, 210677, 216221, 239117, 250997, 378221, 416021, 455621, 549077, 576077, 594437, 680621

Offset: 1

Reinhard Zumkeller, Aug 12 2008

nonn

Intersection of A143203 and A001358.

Sum_{n>=2} 1/a(n) > 0.02187310784. - R. J. Mathar, Jan 23 2013

			a(1) = 3*7 = 3*(3+4) = 21;
a(2) = 7*11 = 7*(7+4) = 77;
a(3) = 13*17 = 13*(13+4) = 221;
a(4) = 19*23 = 19*(19+4) = 437.

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

Cf. A037074, A111192.

a143206 n = a143206_list !! (n-1)
a143206_list = (3*7) : f a000040_list where
   f (p:ps@(p':_)) | p'-p == 4 = (p*p') : f ps
                   | otherwise = f ps
-- Reinhard Zumkeller, Sep 13 2011

[(p*(p+4)): p in PrimesUpTo(1000)| IsPrime(p+4)]; // Vincenzo Librandi, Jan 04 2018

fQ[n_] := Block[{fi = FactorInteger@ n}, Last@# & /@ fi == {1, 1} && Differences[ First@# & /@ fi] == {4}]; Select[ Range@ 700000, fQ] (* Robert G. Wilson v, Feb 08 2012 *)

lista(nn) = forprime(p=2, nn, if (isprime(q=p+4), print1(p*q, ", "))); \\ Michel Marcus, Jan 04 2018

a(n) = A023200(n)*A046132(n).

A160440 Smaller member of a pair (p,q) of cousin primes such that p and q are in different centuries.

Original entry on oeis.org

97, 397, 499, 1297, 1597, 1999, 2797, 3697, 4999, 6199, 6997, 7699, 9199, 10099, 10597, 12097, 13099, 16699, 18397, 20899, 21397, 21499, 21799, 23197, 23599, 25999, 26497, 27697, 27799, 27997, 32299, 32797, 33199, 34297, 35797, 38197, 38299, 39499, 42697

Offset: 1

Ki Punches, May 13 2009

nonn

Sequence is probably infinite.
Dickson's conjecture implies there are infinitely many pairs of primes (100*k-3, 100*k+1) and infinitely many pairs of primes (100*k-1, 100*k+3). - Robert Israel, Mar 28 2023
It appears that every integer occurs as the difference round((a(n+1)-a(n))/100); all numbers 1..298 occur as these differences for a(n) < 1000000000. - Hartmut F. W. Hoft, May 18 2017

			Cousin primes 1597 and 1601 are in successive (that is 16th and 17th) centuries.

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

Cf. A046132, A160370

R:= NULL: count:= 0:
for i from 1 while count < 100 do
  if ((i mod 3 = 1) and isprime(100*i-3) and isprime(100*i+1)) then
     R:= R, 100*i-3; count:= count+1
  elif ((i mod 3 = 2) and isprime(100*i-1) and isprime(100*i+3)) then
     R:= R, 100*i-1; count:= count+1
fi od:
R; # Robert Israel, Mar 28 2023

a160440[n_] := Map[Last, Select[Map[{NextPrime[#, 1], NextPrime[#, -1]}&, Range[100, n, 100]], First[#]-Last[#]==4&]]
a160440[43000] (* data *) (* Hartmut F. W. Hoft, May 18 2017 *)

{A023200(n): [A023200(n)/100] <> [A046132(n)/100]}, where [..]=floor(..).

Edited by R. J. Mathar, May 14 2009

A178071 Numbers k such that exactly one d, 2 <= d <= k/2, exists which divides binomial(k-d-1, d-1) and is not coprime to k.

Original entry on oeis.org

14, 16, 18, 22, 27, 28, 39, 55, 65, 77, 85, 221, 437, 1517, 2021, 4757, 6557, 9797, 11021, 12317, 16637, 27221, 38021, 50621, 53357, 77837, 95477, 99221, 123197, 145157, 159197, 194477, 210677, 216221, 239117, 250997, 378221, 416021, 455621, 549077, 576077

Offset: 1

Vladimir Shevelev, May 19 2010

nonn

Note that every d > 1 divides binomial(k-d-1, d-1), if gcd(k,d)=1.
As shown in the Shevelev link, the sequence contains p*(p+4) for every p >= 7 in A023200.  Thus it is infinite if A023200 is infinite. - Robert Israel, Feb 18 2016
Moreover, similar to proof of Theorem 1 in this link, one can prove that a number m > 85 is a member if and only if it has such a form. - Vladimir Shevelev, Feb 23 2016

Robert Price, Table of n, a(n) for n = 1..179
R. J. Mathar, Corrigendum to "On the divisibility of...", arXiv:1109.0922 [math.NT], 2011.
Vladimir Shevelev, On divisibility of binomial(n-i-1,i-1) by i, Intl. J. of Number Theory 3, no.1 (2007), 119-139.

Cf. A001359, A138389, A178101.

filter:= proc(n) local d, b,count;
  count:= 0;
  b:= 1;
  for d from 2 to n/2 do
     b:= b * (n-2*d+1)*(n-2*d+2)/(n-d)/(d-1);
     if igcd(d,n) <> 1 and b mod d = 0 then
        count:= count+1;
        if count = 2 then return false fi;
     fi
  od;
  evalb(count=1);
end proc:
select(filter, [$1..10^4]); # Robert Israel, Feb 17 2016

Select[Range@ 4000, Function[n, Count[Range[n/2], k_ /; And[! CoprimeQ[n, k], Divisible[Binomial[n - k - 1, k - 1], k]]] == 1]] (* Michael De Vlieger, Feb 17 2016 *)

isok(n) = {my(nb = 0); for (d=2, n\2, if ((gcd(d, n) != 1) && ((binomial(n-d-1,d-1) % d) == 0), nb++); if (nb > 1, return (0));); nb == 1;} \\ Michel Marcus, Feb 17 2016

{k: A178101(k) = 1}.

a(15)-a(23) from Michel Marcus, Feb 17 2016

a(24)-a(41) (from theorem in the Shevelev link) from Robert Price, May 14 2019

A015915 Numbers k such that sigma(k) + 8 = sigma(k+8).

Original entry on oeis.org

3, 5, 11, 23, 27, 29, 53, 59, 71, 89, 101, 131, 149, 173, 191, 233, 263, 269, 359, 389, 401, 431, 449, 479, 491, 563, 569, 593, 599, 653, 683, 701, 719, 743, 761, 821, 911, 929, 983, 1013, 1031, 1061, 1109, 1163, 1193, 1223, 1229, 1283, 1289

Offset: 1

Robert G. Wilson v

nonn

Different from A023202. Below 1000000 four composites were found [27, 1615, 1885, 218984] satisfying the "sigma(k) + 8 = sigma(k+8)" relation, together with more than 8000 primes. - Labos Elemer, May 23 2000

			sigma(27) + 8 = 48 = sigma(27+8), so 27 is in the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000203, A015913-A015917, A023200-A023203, A046133, A001359, A054799.

Composite solutions are in A059118.

Select[Range[1300],DivisorSigma[1,#]+8==DivisorSigma[1,#+8]&] (* Harvey P. Dale, Jul 16 2011 *)

is(n)=sigma(n)+8==sigma(n+8) \\ Charles R Greathouse IV, Mar 11 2014

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A049489 Primes p such that p + 32 is also prime.

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A049490 a(n) and a(n)+64 both prime.

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A094343 List of pairs of primes (p, q) with q - p = 4.

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A156104 Primes p such that p+36 is also prime.

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Magma

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A218867 Number of prime pairs {p,q} with p>q and {p-4,q+4} also prime such that p+(1+(n mod 6))q=n if n is not congruent to 4 (mod 6), and p-q=n and q

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

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Mathematica

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A143206 Product of the n-th cousin prime pair.

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