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A117563 a(n) = A118534(n)/A117078(n) unless A117078(n) = 0 in which case a(n) = 0. 78
0, 0, 1, 0, 3, 1, 5, 3, 1, 9, 1, 3, 13, 3, 1, 1, 19, 5, 9, 23, 1, 15, 11, 9, 3, 33, 11, 35, 21, 3, 3, 5, 45, 3, 49, 5, 1, 3, 23, 1, 59, 9, 63, 27, 65, 11, 1, 3, 75, 45, 1, 79, 21, 35, 1, 1, 89, 5, 39, 93, 21, 9, 3, 103, 3, 3, 25, 3, 115, 69, 1, 39, 19, 1, 75, 29, 3, 3, 3, 21, 139, 3, 143, 61, 87 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

a(n) is the "level" of prime(n).

There is a unique decomposition of the primes: provided the level a(n) is > 0, we have prime(n) = weight * level + gap, or A000040(n)=A117078(n)*a(n)+A001223(n).

a(n) = 0 only for primes 2, 3 and 7.

A118534(n) = prime(n) - g(n) or A000040(n) - A001223(n) if prime(n) - g(n) > g(n), 0 otherwise.

LINKS

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

Remi Eismann, Java program to decompose a prime as weight*level + gap, or 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], 2007-2010.

EXAMPLE

a(7)=15/3=5; a(14)=39/13=3; a(16)=47/47=1; a(18)=55/11=5; a(29)=105/5=11.

MATHEMATICA

a34[n_] := If[n == 1 || n == 2 || n == 4, 0, 2 Prime[n] - Prime[n+1]];

a78[n_] := Block[{a, p = Prime[n], np = Prime[n+1]}, a = Min[Select[ Divisors[2p - np], # > np - p& ]]; If[a == Infinity, 0, a]];

a[n_] := If[a78[n] == 0, 0, a34[n]/a78[n]];

Array[a, 85] (* Jean-François Alcover, Nov 02 2018, after Robert G. Wilson v in A118534  *)

CROSSREFS

Cf. A117078, A118534.

Sequence in context: A102662 A350279 A142048 * A060439 A206283 A135224

Adjacent sequences:  A117560 A117561 A117562 * A117564 A117565 A117566

KEYWORD

nonn

AUTHOR

Rémi Eismann, Apr 29 2006, Feb 14 2008

EXTENSIONS

More terms from Robert G. Wilson v, May 05 2006

Edited by N. J. A. Sloane, May 14 2006

STATUS

approved

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Last modified September 26 20:57 EDT 2022. Contains 357050 sequences. (Running on oeis4.)