A117078 a(n) is the smallest k such that prime(n+1) = prime(n) + (prime(n) mod k), or 0 if no such k exists.

0, 0, 3, 0, 3, 9, 3, 5, 17, 3, 25, 11, 3, 13, 41, 47, 3, 11, 7, 3, 67, 5, 7, 9, 31, 3, 9, 3, 5, 33, 41, 25, 3, 43, 3, 29, 151, 53, 7, 167, 3, 19, 3, 7, 3, 17, 199, 73, 3, 5, 227, 3, 11, 7, 251, 257, 3, 53, 7, 3, 13, 31, 101, 3, 103, 101, 13, 109, 3, 5, 347, 9, 19, 367, 5, 13, 127, 131, 131, 19, 3

Offset

1,3

Comments

There is a unique decomposition of the primes: provided the weight a(n) is > 0, we have prime(n) = weight * level + gap, or A000040(n)=a(n)*A117563(n)+A001223(n).

a(n) is the smallest divisor of A118534(n) greater than A001223(n) (gap).

a(n) == 0 (mod 2) only for n = {1, 2 or 4}. - Robert G. Wilson v, May 05 2006

a(n) = 0 only for primes 2, 3 and 7. Conjecture: 2, 3 and 7 are the only primes for which log(A000040(n)) < sqrt(A001223(n)).

a(n) > 0 if and only if 2*prime(n+1) < 3*prime(n). - Thomas Ordowski, Nov 25 2013

Links

Remi Eismann, Table of n, a(n) for n = 1..10000

Rémi Eismann, Decomposition into weight * level + jump and application to a new classification of primes, arXiv:0711.0865 [math.NT], 2007-2010.

Fabien Sibenaler, Program in assembly that gives the decomposition of a prime number [prime = weight * level + gap, or A000040(n) = A117078(n) * A117563(n) + A001223(n)]

Example

For n = 1 we have prime(n) = 2, prime(n+1) = 3; there is no k such that 3 - 2 = 1 = (2 mod k), hence a(1) = 0.
For n = 3 we have prime(n) = 5, prime(n+1) = 7; 3 is the smallest k such that 7 - 5 = 2 = (5 mod k), hence a(3) = 3.
For n = 19 we have prime(n) = 67, prime(n+1) = 71; 7 is the smallest k such that 71 - 67 = 4 = (67 mod k), hence a(19) = 7.

Mathematica

f[n_] := Block[{a, p = Prime@n, np = Prime[n + 1]}, a = Min@ Select[ Divisors[2p - np], # > np - p &]; If[a == Infinity, 0, a]]; Array[f, 80] (* Robert G. Wilson v, May 08 2006 *)

Prog

(PARI) {m=78; for(n=1, m, p=prime(n); d=prime(n+1)-p; k=0; j=1; while(k==0&&j<p, if(p%j!=d, j++, k=j)); print1(k, ", "))}

Crossrefs

Cf. A074822 (k=5), A118534, A117563.

Sequence in context: A137339 A230184 A132330 * A021333 A348670 A104141

Adjacent sequences: A117075 A117076 A117077 * A117079 A117080 A117081

Keyword

nonn

Author

Rémi Eismann, Apr 18 2006, Dec 10 2006, Feb 14 2008

Extensions

Edited and corrected by Don Reble and Klaus Brockhaus, Apr 21 2006

Status

approved