A007508 -id:A007508 - cp's OEIS frontend

A080841 Number of pairs (p,q) of (not necessarily consecutive) primes with q-p = 6 and q < 10^n.

0, 15, 74, 411, 2447, 16386, 117207, 879908, 6849047, 54818296, 448725003, 3741217498

Offset: 1

Jason Earls, Mar 28 2003

nonn

Note that one has to be careful to distinguish between pairs of consecutive primes (p,q) with q-p = 6 (A031924), and pairs of primes (p,q) with q-p = 6 (A023201). Here we consider the latter, whereas A093738 considers the former. - N. J. A. Sloane, Mar 07 2021

A. Granville, G. Martin, Prime number races, Amer. Math. Monthly vol 113, no 1 (2006) p 1.
Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021].

Cf. A023201, A046117, A007508, A080840, A006512, A046132, A046117.

A093737 Number of prime pairs below 10^n having a difference of 4.

Original entry on oeis.org

0, 7, 40, 202, 1215, 8143, 58621, 440257, 3424679, 27409998, 224373160, 1870585458, 15834656002, 135779962759, 1177207270203, 10304191320776, 90948823579814, 808675898548205

Offset: 1

Enoch Haga, Apr 15 2004

			a(2) = 7 because there are 7 prime gaps of 4 below 10^2.

Siegfried "Zig" Herzog, Frequency of Occurrence of Prime Gaps
T. Oliveira e Silva, S. Herzog, and S. Pardi, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4.10^18, Math. Comp., 83 (2014), 2033-2060.

Cf. A007508, A093738.

a(n)=my(p=2,s); forprime(q=3,10^n, if(q-p==4, s++); p=q); s \\ Charles R Greathouse IV, Feb 05 2016

20 N=1:dim T(34); 30 A=nxtprm(N); 40 N=A; 50 B=nxtprm(N); 60 D=B-A; 70 for x=2 to 34 step 2; 80 if D=X and B<10^2+1 then T(X)=T(X)+1; 90 next X; 100 if B>10^2+1 then 140; 110 B=A; 120 N=N+1; 130 goto 30; 140 for x=2 to 34 step 2; 150 print T(X);, 160 next (This program simultaneously finds values from 2 to 34 -- if gap=2 add 1-- adjust lines 80 and 100 for desired 10^n)

a(10)-a(13) from Washington Bomfim, Jun 22 2012

a(14)-a(18) from S. Herzog's website added by Giovanni Resta, Aug 14 2018

A093972 Number of prime pairs below 10^n having a difference of 36.

Original entry on oeis.org

0, 0, 0, 1, 54, 767, 10194, 114028, 1171524, 11593345, 112117864, 1069300358, 10109375268, 95086492066, 891766724342, 8351325770872, 78171510043913, 731828924318516

Offset: 1

Enoch Haga, Apr 24 2004

			a(4) = 1 because there is 1 prime gap of 36 below 10^4.

Siegfried "Zig" Herzog, Frequency of Occurrence of Prime Gaps
T. Oliveira e Silva, S. Herzog, and S. Pardi, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4.10^18, Math. Comp., 83 (2014), 2033-2060.

Cf. A007508, A093752, A093973.

a(10)-a(13) from Washington Bomfim, Jun 22 2012

a(14)-a(18) from S. Herzog's website added by Giovanni Resta, Aug 14 2018

A118552 Sum of the twin prime pairs less than 10^n.

Original entry on oeis.org

20, 488, 24236, 1726412, 109114568, 7424366648, 545678596592, 41205774636932, 3234489739234676, 260643410442091112, 21446976192435396140, 1795640886305783918948, 152542601906447626814216, 13119246582832293524505360

Offset: 1

Cino Hilliard, May 07 2006

The PARI program can compute the first 9 terms in reasonable time. a(10) was computed by the program in the link. This took 145 sec on a p4 2.53 GHz processor while a(13) took 1.4 days and a(14) took 15 days with multitasking. The sum of twin primes < 10^n divided by 4 gives a very good approximation for the number of twin primes < 10^(2n). E.g., sum of twin primes <= 10^8 divided by 4 = 10301443659233. Pi_2(10^16) = 10304185697298. This is an error of 0.00002661. Pi_2(n): Number of twin prime pairs <= n.

			(3,5),(5,7) are the two twin prime pairs less than 10. These add up to 20, the first term in the sequence.

Cino Hilliard, Sum of twin primes less than 10^n. [Broken link]

Cf. A007508.

sumtwins(n) = { local(x,j,s,sr,p10x); for(x=1,n, s=0; p10x=10^x; forstep(j=3,10^x,2, if(j+2 < p10x && ispseudoprime(j) && ispseudoprime(j+2),s+=j+j+2); ); print1(s","); ) }

2 more terms from Giovanni Resta, May 08 2006

a(13) and a(14) added, comment expanded, program at link improved, and example edited by Cino Hilliard, Nov 18 2008

A181678 Number of twin Ramanujan prime pairs less than 10^n.

Original entry on oeis.org

0, 0, 10, 73, 508, 3468, 25629, 194614, 1537504, 12447679, 102834428

Offset: 1

T. D. Noe, Nov 18 2010

J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly 116 (2009) 630-635.
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2

Cf. A178127 (lesser of twin Ramanujan primes), A007508 (number of twin primes pairs < 10^n), A190654 (twin Ramanujan primes).

nn=50000; t=Table[0,{nn}]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s

use ntheory ":all"; for my $n (0..9) { my $r = ramanujan_primes(10**$n); my $t=0; for (0..$#$r-1) { $t++ if $r->[$+1] - $r->[$] == 2; } say "$n $t"; } # Dana Jacobsen, Sep 06 2015

a(10)-a(11) from Dana Jacobsen, Apr 29 2015

A093683 Number of pairs of twin primes <= 10^n-th prime.

Original entry on oeis.org

4, 25, 174, 1270, 10250, 86027, 738597, 6497407, 58047180, 524733511, 4789919653, 44073509102, 408231310520

Offset: 1

Enoch Haga, Apr 09 2004

This sequence is >= the values of pi(10^n): 4, 25, 168, 1229, ... in A006880.

a(0) = 0. - Eduard Roure Perdices, Dec 23 2022

			a(1) = 4 because there are 4 twin primes <= 29, the 10th prime: (3,5), (5,7), (11,13), and (17,19). (29,31) is not counted because it is not entirely <= 29.

Enoch Haga, "Wandering through a prime number desert," Table 6, in Exploring prime numbers on your PC and the Internet, 2001 (ISBN 1-885794-17-7).

Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and Stephan Ramon Garcia, The Bateman-Horn Conjecture: Heuristics, History, and Applications, arXiv:1807.08899 [math.NT], 2018-2019. See Table 5 p. 40.
Thomas R. Nicely, Twin prime count.
Index entries for sequences related to numbers of primes in various ranges

See A049035 for another version. - R. J. Mathar, Sep 05 2008

Cf. A006880, A007508.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; c = 0; p = q = 1; Do[l = Prime[10^n]; While[q <= l, If[p + 2 == q, c++ ]; p = q; q = NextPrim[p]]; Print[c], {n, 12}] (* Robert G. Wilson v, Apr 10 2004 *)

from sympy import prime, sieve # use primerange for larger terms
def afind(terms):
  c, prevp = 0, 1
  for n in range(1, terms+1):
    for p in sieve.primerange(prevp+1, prime(10**n)+1):
      if prevp == p - 2: c += 1
      prevp = p
    print(c, end=", ")
afind(6) # Michael S. Branicky, Apr 25 2021

Count twin primes <= p_{10^n}: 10th prime, 100th prime, etc.

a(9) from Michael S. Branicky, Apr 25 2021
a(10) from Eduard Roure Perdices, May 08 2021
a(11) from Eduard Roure Perdices, Feb 03 2022
a(12) from Eduard Roure Perdices, Dec 23 2022
a(13) from Eduard Roure Perdices, Jan 24 2024

A093739 Number of prime pairs below 10^n having a difference of 8.

Original entry on oeis.org

0, 1, 15, 101, 773, 5569, 42352, 334180, 2695109, 22160841, 185402143, 1573331564, 13515180171, 117333792953, 1028087693781, 9081524454631, 80799078096971, 723494891844589

Offset: 1

Enoch Haga, Apr 15 2004

			a(3) = 15 because there are 15 prime gaps of 8 below 10^3.

Siegfried "Zig" Herzog, Frequency of Occurrence of Prime Gaps
T. Oliveira e Silva, S. Herzog, and S. Pardi, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4.10^18, Math. Comp., 83 (2014), 2033-2060.

Cf. A007508, A093738, A093740.

20 N=1:dim T(34);
30 A=nxtprm(N);
40 N=A;
50 B=nxtprm(N);
60 D=B-A;
70 for x=2 to 34 step 2;
80 if D=X and B<10^2+1 then T(X)=T(X)+1;
90 next X;
100 if B>10^2+1 then 140;
110 B=A;
120 N=N+1;
130 goto 30;
140 for x=2 to 34 step 2;
150 print T(X);,
160 next
## (This program simultaneously finds values from 2 to 34 - if gap=2 add 1- adjust lines 80 and 100 for desired 10^n)

a(10)-a(13) from Washington Bomfim, Jun 20 2012

a(14)-a(18) from S. Herzog's website added by Giovanni Resta, Aug 14 2018

A093740 Number of prime pairs below 10^n having a difference of 10.

Original entry on oeis.org

0, 0, 16, 119, 916, 7079, 54431, 430016, 3484767, 28764495, 241298621, 2052293026, 17663498098, 153590992984, 1347587381486, 11917605558274, 106139298948562, 951243890034661

Offset: 1

Enoch Haga, Apr 15 2004

			a(3) = 16 because there are 16 prime gaps of 10 below 10^3.

Siegfried "Zig" Herzog, Frequency of Occurrence of Prime Gaps
T. Oliveira e Silva, S. Herzog, and S. Pardi, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4.10^18, Math. Comp., 83 (2014), 2033-2060.

Cf. A007508, A093739, A093741.

20 N=1:dim T(34);
30 A=nxtprm(N);
40 N=A;
50 B=nxtprm(N);
60 D=B-A;
70 for x=2 to 34 step 2;
80 if D=X and B<10^2+1 then T(X)=T(X)+1;
90 next X;
100 if B>10^2+1 then 140;
110 B=A;
120 N=N+1;
130 goto 30;
140 for x=2 to 34 step 2;
150 print T(X);,
160 next
## (This program simultaneously finds values from 2 to 34 - if gap=2 add 1- adjust lines 80 and 100 for desired 10^n)

a(10)-a(13) from Washington Bomfim, Jun 20 2012

a(14)-a(18) from S. Herzog's website added by Giovanni Resta, Aug 14 2018

A093741 Number of prime pairs below 10^n having a difference of 12.

Original entry on oeis.org

0, 0, 7, 105, 964, 8005, 65513, 538382, 4468957, 37588207, 319972455, 2753597777, 23931289984, 209796541197, 1853552707266, 16490531511496, 147632682729513, 1329180643501191

Offset: 1

Enoch Haga, Apr 15 2004

			a(3) = 7 because there are 7 prime gaps of 12 below 10^3.

Siegfried "Zig" Herzog, Frequency of Occurrence of Prime Gaps
T. Oliveira e Silva, S. Herzog, and S. Pardi, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4.10^18, Math. Comp., 83 (2014), 2033-2060.

Cf. A007508, A093740, A093742.

Table[Count[Differences[Prime[Range[PrimePi[10^n]]]],12],{n,10}] (* The program generates the first 10 terms of the sequence. *) (* Harvey P. Dale, Jun 24 2025 *)

20 N=1:dim T(34); 30 A=nxtprm(N); 40 N=A; 50 B=nxtprm(N); 60 D=B-A; 70 for x=2 to 34 step 2; 80 if D=X and B<10^2+1 then T(X)=T(X)+1; 90 next X; 100 if B>10^2+1 then 140; 110 B=A; 120 N=N+1; 130 goto 30; 140 for x=2 to 34 step 2; 150 print T(X);, 160 next (This program simultaneously finds values from 2 to 34 -- if gap=2 add 1-- adjust lines 80 and 100 for desired 10^n)

a(10)-a(13) from Washington Bomfim, Jun 22 2012

a(14)-a(18) from S. Herzog's website added by Giovanni Resta, Aug 14 2018

A093742 Number of prime pairs below 10^n having a difference of 14.

Original entry on oeis.org

0, 0, 7, 54, 484, 4233, 35394, 293201, 2464565, 20943953, 179718000, 1556469349, 13597962107, 119731244640, 1061769557793, 9476573902533, 85076550195696, 767846949916102

Offset: 1

Enoch Haga, Apr 15 2004

			a(3) = 7 because there are 7 prime gaps of 14 below 10^3.

Siegfried "Zig" Herzog, Frequency of Occurrence of Prime Gaps
T. Oliveira e Silva, S. Herzog, and S. Pardi, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4.10^18, Math. Comp., 83 (2014), 2033-2060.

Cf. A007508, A093741, A093743.

20 N=1:dim T(34); 30 A=nxtprm(N); 40 N=A; 50 B=nxtprm(N); 60 D=B-A; 70 for x=2 to 34 step 2; 80 if D=X and B<10^2+1 then T(X)=T(X)+1; 90 next X; 100 if B>10^2+1 then 140; 110 B=A; 120 N=N+1; 130 goto 30; 140 for x=2 to 34 step 2; 150 print T(X);, 160 next (This program simultaneously finds values from 2 to 34 -- if gap=2 add 1-- adjust lines 80 and 100 for desired 10^n)

a(10)-a(13) from Washington Bomfim, Jun 22 2012

a(14)-a(18) from S. Herzog's website added by Giovanni Resta, Aug 14 2018

cp's OEIS Frontend

A080841 Number of pairs (p,q) of (not necessarily consecutive) primes with q-p = 6 and q < 10^n.

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PARI

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A093737 Number of prime pairs below 10^n having a difference of 4.

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PARI

UBASIC

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A093972 Number of prime pairs below 10^n having a difference of 36.

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A118552 Sum of the twin prime pairs less than 10^n.

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PARI

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A181678 Number of twin Ramanujan prime pairs less than 10^n.

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Mathematica

Perl

Extensions

A093683 Number of pairs of twin primes <= 10^n-th prime.

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Mathematica

Python

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A093739 Number of prime pairs below 10^n having a difference of 8.

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UBASIC

Extensions

A093740 Number of prime pairs below 10^n having a difference of 10.

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