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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. 77
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; text; internal format)
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
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
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

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Last modified May 29 08:39 EDT 2024. Contains 372926 sequences. (Running on oeis4.)