



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
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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], 20072010.


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] (* JeanFrançois Alcover, Nov 02 2018, after Robert G. Wilson v in A118534 *)


CROSSREFS

Cf. A117078, A118534.
Sequence in context: A122510 A102662 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



