A031172 a(n) = prime(n+10) - prime(n).

29, 34, 36, 36, 36, 40, 42, 42, 44, 42, 42, 42, 42, 46, 50, 48, 44, 46, 42, 42, 54, 52, 54, 50, 52, 50, 54, 56, 58, 60, 52, 50, 54, 54, 48, 48, 54, 60, 60, 56, 54, 58, 50, 58, 60, 64, 58, 48, 50, 52, 50, 54, 66, 60, 56, 54, 62, 66, 70, 68, 70, 66, 60, 62, 66, 66, 58

Offset: 1

Author

Jeff Burch

Keywords

• nonn
• changed

Comments

In principle, moderate values should appear infinitely many times, by analogy with twin primes hypothesis. For example, a(n) = 44 for n = 9, 17, 206, 1604467, 12905293, 18008874, 26545460, 32655424, 57848470, 58313630, 59022635, 66275281, 81581956, 123780499, 160884754, 167797255, 179786560, 181569324, 239542290, ... - Zak Seidov, Sep 14 2014, edited by M. F. Hasler, Dec 03 2018

According to the k-tuple conjecture, any admissible k-tuple of primes occurs with calculable nonzero asymptotic density, i.e., in particular, infinitely many times. For k = 11, number of primes in the interval [prime(n), prime(n+10)], the smallest possible diameter of a k-tuple is A008407(11) = 36, and there are A083409(11) = 2 such constellations: {0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36}, first occurring at A213646(1) = 1418575498573, and {0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36}, first occurring at A213647(1) = 11. The combined list { prime(n) | a(n) = 36 } is A257129. - M. F. Hasler, Dec 03 2018

Links

• Zak Seidov, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040.

Cf. A001223, A031131, A031165, A031166, A031167, A031168, A031169, A031170, A031171.

Cf. A008407, A083409, A213646, A213647, A257129.

Programs

GAP

P:=Filtered([1..400],IsPrime);; a:=List([1..Length(P)-10],n->P[n+10]-P[n]); # Muniru A Asiru, Dec 06 2018

Haskell

a031172_list = zipWith (-) (drop 10 a000040_list) a000040_list
a031172 n = a031172_list !! (n-1)  -- Reinhard Zumkeller, Aug 23 2015

Magma

[NthPrime(n+10)-NthPrime(n): n in [1..100] ]; // Vincenzo Librandi, Apr 23 2011

Maple

A031172:=n->ithprime(n+10)-ithprime(n): seq(A031172(n), n=1..50);

Mathematica

Table[Prime[n + 10] - Prime[n], {n, 50}] (* Wesley Ivan Hurt, Sep 14 2014 *)

PARI

A031172(n)=prime(n+10)-prime(n) \\ M. F. Hasler, Dec 03 2018

Python

from sympy import prime
for n in range(1,100): print(prime(n+10)-prime(n)) # Stefano Spezia, Dec 06 2018

Sage

[(nth_prime(n+10) - nth_prime(n)) for n in (1..100)] # G. C. Greubel, Dec 04 2018

Formula

a(n) = A000040(n+10) - A000040(n). - Wesley Ivan Hurt, Sep 14 2014

Extensions

Offset changed from 2 to 1; added a(1)=29 by Vincenzo Librandi, Apr 23 2011