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A117078
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a(n) = smallest k such that prime(n+1) = prime(n) + (prime(n) mod k), or 0 if no such k exists.
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74
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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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 ln(A000040(n)) < SQRT(A001223(n)).
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LINKS
| Remi Eismann, Table of n, a(n) for n = 1..10000
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)]
Rémi Eismann, Decomposition into weight * level + jump and application to a new classification of primes, arXiv:0711.0865 [math.NT]
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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.
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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] (from Robert G. Wilson v (rgwv(at)rgwv.com), May 08 2006)
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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, ", "))}
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CROSSREFS
| Cf. A118534, A117563.
Sequence in context: A099093 A137339 A132330 * A021333 A104141 A060533
Adjacent sequences: A117075 A117076 A117077 * A117079 A117080 A117081
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KEYWORD
| nonn
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AUTHOR
| Remi Eismann (reismann(AT)free.fr), Apr 18 2006, Dec 10 2006, Feb 14 2008
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EXTENSIONS
| Edited and corrected by Don Reble (djr(AT)nk.ca) and Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Apr 21 2006
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