cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A152052 Number of cousin primes < 10^n.

Original entry on oeis.org

2, 9, 41, 203, 1216, 8144, 58622, 440258, 3424680, 27409999, 224373161, 1870585459, 15834656003, 135779962760, 1177207270204
Offset: 1

Views

Author

Cino Hilliard, Nov 22 2008

Keywords

Comments

The convention here is that only the lower member of a cousin prime pair be less than the selected bound 10^n.
Cousin primes, like twin primes, can be approximated by the Hardy-Littlewood formula for the number of twin primes < n. For example, the number of cousin primes < 10^12 = 1870585459 while Hardy-Littlewood gives 1870559867. The sum of cousin primes < 10^6 divided by 4 also approximates the number of cousin primes < 10^12 with 1844802199. These two methods are asymptotic to the true value as n -> infinity.

Examples

			(3,7) and (7,11) are cousin primes < 10 since 7 < 10. So 2 is the first entry in the sequence.

Links

Crossrefs

Variant of A080840. [R. J. Mathar, Nov 27 2008]
Cf. A152127, A333587.

Formula

A cousin prime pair is a pair of primes that differ by 4.

Extensions

a(13)-a(15) from Martin Ehrenstein, Sep 03 2021

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

Original entry on oeis.org

0, 15, 74, 411, 2447, 16386, 117207, 879908, 6849047, 54818296, 448725003, 3741217498
Offset: 1

Views

Author

Jason Earls, Mar 28 2003

Keywords

Comments

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

Links

  • 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].

Crossrefs

Cf. A023201, A046117, A007508, A080840, A006512, A046132, A046117.
See also A031924, A093738.

Programs

  • PARI
    {c=0; p=7; for(n=1,9, while(p<10^n,if(isprime(p-6),c++); p=nextprime(p+1)); print1(c,","))}

Extensions

a(8) and a(9) from Klaus Brockhaus, Mar 30 2003
More terms from R. J. Mathar, Aug 05 2007

A152127 Sum of cousin primes < 10^n.

Original entry on oeis.org

28, 766, 34522, 1706602, 107863996, 7379208796, 542100094312, 41248685420836, 3233516261489332, 260607555289408894, 21446383929686290726, 1795656778320649469818, 152541729206365604807782
Offset: 1

Views

Author

Cino Hilliard, Nov 25 2008

Keywords

Comments

Cousin primes are prime pairs that differ by 4. The convention here is to count a cousin pair as long as the first cousin of the pair is less than or equal to the specified bound which in this sequence is 10^n.
"According to the first Hardy-Littlewood conjecture, the cousin primes have the same asymptotic density as the twin primes." See the MathWorld link. The (sum of cousin prime pairs < 10^n)/4 ~ number of cousin primes < 10^2n.

Examples

			(3,7) and (7,11) are the cousin primes < 10. These add up to 28 the first entry in the sequence.

Links

Crossrefs

Cf. A023200, A046132, A080840, A152052.

Programs

  • PARI
    lista(pmax) = {my(sm = 10, prev = 2, pow = 10); forprime(p = 3, pmax, if(p == prev + 4, sm += (prev + p)); if(p > pow, print1(sm, ", "); pow *= 10); prev = p);} \\ Amiram Eldar, Jul 06 2024

Extensions

Data corrected by Amiram Eldar, Jul 06 2024
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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