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%I #32 Apr 01 2020 12:18:17
%S 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,
%T 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,
%U 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
%N a(n) is the smallest k such that prime(n+1) = prime(n) + (prime(n) mod k), or 0 if no such k exists.
%C 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 a(n) is the smallest divisor of A118534(n) greater than A001223(n) (gap).
%C a(n) == 0 (mod 2) only for n = {1, 2 or 4}. - _Robert G. Wilson v_, May 05 2006
%C 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)).
%C a(n) > 0 if and only if 2*prime(n+1) < 3*prime(n). - _Thomas Ordowski_, Nov 25 2013
%H Remi Eismann, <a href="/A117078/b117078.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémi Eismann, <a href="http://arXiv.org/abs/0711.0865">Decomposition into weight * level + jump and application to a new classification of primes</a>, arXiv:0711.0865 [math.NT], 2007-2010.
%H 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)]
%e 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 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 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 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 *)
%o (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 Cf. A074822 (k=5), A118534, A117563.
%K nonn
%O 1,3
%A _Rémi Eismann_, Apr 18 2006, Dec 10 2006, Feb 14 2008
%E Edited and corrected by _Don Reble_ and _Klaus Brockhaus_, Apr 21 2006
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