1,1

These are the [first] primes which cannot be part of a gb-sequence (except as seed).

Is this sequence finite or infinite ?

GNU-GMP was used to generate the sequence.

[Editor's note: This probably refers not to this sequence but to the "gb-sequences" themselves, e.g. the one starting with 4499221 reaches a peak of approximately 10^110. - M.H.]

Comments from M. F. Hasler, Mar 27 2008 (Start) The function f(p)=p/2 if p=2 (mod 3), f(p)=2p else, yields a half-integer for primes p=6k-1, and an even number for primes p=6k+1; in all cases nextprime(f(p)) is defined without ambiguity.

This sequence lists primes p' not in the range of the map p->nextprime(f(p)) (defined on the primes).

Equivalently, p' is listed iff: (1) no even number between p' and the next lower prime is of the form 2p with p=0 or p=1 (mod 3), AND (2) no half-integer between p' and the next lower prime is of the form p/2 with p=2 (mod 3), and p prime (in both conditions).

This characterization allows us to compute the sequence easily (cf. PARI code). - M. F. Hasler, Mar 27 2008

Experimentally, it does not appear that this sequence is be finite. Instead, its (local) density within the primes seems to increase, from roughly 25% for the first terms to nearly 50% at 10^30. The average density seems to remain stable around 40%. - M. F. Hasler, Mar 27 2008

Communication paper by Georges Brougnard.

Georges Brougnard, Definition of GB-sequences.

Georges Brougnard, GB-sequence of length 96, obtained for gb[0]=1381.

Georges Brougnard, GB-sequence of length 63337, obtained for gb[0]=4499221.

A124123 = complement of A007918( A138750( A000040 )) = nextprime( f( { primes } ))

Recurrence for a gb-sequence starting with gb[0] = a prime > 2 (the seed):

| If gb[n] = 2 (mod 3) Then gb[n+1] := least prime > gb[n]/2,

| Else gb[n+1] := least prime > gb[n]*2.

A gb-sequence of length L ends in the loop 7, 17, 11, 7,.. ; gb[L-1]=7.

Example: a(1) = 5 because there is no prime gb[n] such that gb[n+1] = 5.

(PARI) forprime( p=3, 10^3, { for( i=precprime(p-1)+1, p, (2*i)%3==0 & isprime(2*i-1) & next(2); i%2==0 & ( i/2 )%3!=2 & isprime( i/2 ) & next(2)); print1( p", " )) } \\ - M. F. Hasler, Mar 27 2008

(PARI) nextA124123(p)={ while( p=nextprime(p+1), for( i=precprime(p-1)+1, p, (2*i)%3==0 & isprime(2*i-1) & next(2); i%2==0 & ( i/2 )%3!=2 & isprime( i/2 ) & next(2)); return( p )) } \\ - M. F. Hasler, Mar 27 2008

t=2; vector(200, i, t=nextA124123(t)) \\ 3/5 = 60% of the first 200 terms are in 1+3Z:

t=[0, 0]; vector(#%, i, t[%[i]%3]++); t \\ yields [120, 80]

t=10^11; vector(200, i, t=nextA124123(t)) \\ exactly 1/2 = 50% of these terms are in 1+3Z:

t=[0, 0]; vector(#%, i, t[%[i]%3]++); t \\ yields [100, 100]

t=10^30; vector(200, i, t=nextA124123(t+1)); t-10^30 \\ yields 31773 = distance of 200th term beyond 10^30

t=10^30; vector(200, i, t=nextprime(t+1)); (t-1e30)/% \\ yields 0.52..., approx. local density in the primes. - M. F. Hasler, Mar 27 2008

Cf. A007918, A138750-A138753.

Adjacent sequences: A124120 A124121 A124122 this_sequence A124124 A124125 A124126

Sequence in context: A107179 A092442 A135266 this_sequence A128638 A036630 A102841

easy,nonn

Jacques Tramu (jacques.tramu(AT)echolalie.com), Dec 13 2006

Edited by M. F. Hasler (univ-ag.fr/~mhasler), Mar 27 2008