%I A117078
%S A117078 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,
%T A117078 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,
%U A117078 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
%N A117078 a(n) = smallest k such that prime(n+1) = prime(n) + (prime(n) mod k), 
               or 0 if no such k exists.
%C A117078 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).
%C A117078 a(n) is the smallest divisor of A118534(n) greater than A001223(n) (gap).
%C A117078 a(n) == 0 (mod 2) only for n = {1, 2 or 4}. - Robert G. Wilson v May 
               05 2006.
%C A117078 a(n) = 0 only for primes 2, 3 and 7. Conjecture: 2, 3 and 7 are the only 
               primes for which ln(A000040(n)) < SQRT(A001223(n)).
%H A117078 Remi Eismann, <a href="b117078.txt">Table of n, a(n) for n = 1..10000</
               a>
%H A117078 Fabien Sibenaler, <a href="http://reismann.free.fr/download/class_asm.zip">
               Program in assembly that gives the decomposition of a prime number</
               a> [prime = weight * level + gap, or A000040(n) = A117078(n) * A117563(n) 
               + A001223(n)]
%H A117078 Remi Eismann, <a href="http://arXiv.org/abs/0711.0865">Decomposition 
               of natural numbers into weight * level + jump and application to 
               a new classification of prime numbers</a>
%e A117078 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.
%e A117078 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.
%e A117078 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.
%t A117078 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] (from Robert G. Wilson v (rgwv(at)rgwv.com), May 08 2006)
%o A117078 (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,","))}
%Y A117078 Cf. A118534, A117563.
%Y A117078 Sequence in context: A099093 A137339 A132330 this_sequence A021333 A104141 
               A060533
%Y A117078 Adjacent sequences: A117075 A117076 A117077 this_sequence A117079 A117080 
               A117081
%K A117078 nonn
%O A117078 1,3
%A A117078 Remi Eismann (reismann(AT)free.fr), Apr 18 2006, Dec 10 2006, Feb 14 
               2008
%E A117078 Edited and corrected by Don Reble (djr(AT)nk.ca) and Klaus Brockhaus 
               (klaus-brockhaus(AT)t-online.de), Apr 21 2006