A112037 Go through all of the primes p and for each one, factor p-1 into primes. List the primes in order of their first appearance in the p-1 factorizations.

2, 3, 5, 11, 7, 23, 13, 29, 41, 17, 53, 37, 83, 43, 89, 19, 113, 131, 67, 47, 73, 31, 79, 173, 179, 61, 191, 97, 233, 239, 251, 127, 139, 281, 71, 293, 101, 103, 107, 163, 59, 359, 193, 199, 137, 419, 431, 443, 151, 491, 509, 181, 109, 277, 593, 149, 307, 641, 653

Offset

2,1

Comments

The length of this list of distinct prime factors of p-1 encountered through p=prime(n) is given by A055768. - Ray Chandler, Nov 30 2005

If "p-1" is changed to "p+1" we get A236388. - N. J. A. Sloane, Jan 24 2014

A permutation of the primes by Dirichlet's theorem on arithmetic progressions: for any pair (r,s) of integers such that gcd(r,s)=1 there are infinitely many primes in the sequence r + k*s; choose r=1 and s=p. - Joerg Arndt, Mar 20 2016

Links

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Wikipedia, Dirichlet's theorem on arithmetic progressions

Example

We start with the second prime, 3. 3-1 = 2, so 2 is the first term.
5-1 = 2*2, nothing new.
7-1 = 2*3 and 3 is new, so that is the second term.
11-1 = 2*5 and we get 5; etc.

Mathematica

lst = {}; r[n_] := (len = Length@lst; lst = Flatten@ Join[lst, Select[First /@ FactorInteger[Prime@n - 1], ! MemberQ[lst, # ] &]]; If[l < Length@lst, 1, 0]); Do[ r[n], {n, 214}]; lst (* Robert G. Wilson v, Nov 30 2005 *)
DeleteDuplicates[Rest[Flatten[FactorInteger[#][[All, 1]]&/@ (Prime[ Range[ 250]]-1)]]] (* Harvey P. Dale, May 26 2019 *)

Prog

(PARI) g=1; forprime(p=2, 299, f=factorint(p-1)[, 1]; z=factorback(f); r=z/gcd(z, g); g*=r; if(r>1, print(r, " ", p))); \\ Jack Brennen, Nov 28 2005
(GAP) Set(Flat(List(Filtered([3..1500], IsPrime), i->Factors(i-1)))); # Muniru A Asiru, Dec 06 2018

Crossrefs

Cf. A055768, A112038, A114461, A236388.

Sequence in context: A084333 A288833 A067663 * A087583 A372284 A233098

Adjacent sequences: A112034 A112035 A112036 * A112038 A112039 A112040

Keyword

easy,nonn

Author

Michel Dauchez (mdzdm(AT)yahoo.fr), Nov 28 2005

Extensions

Better description from Jack Brennen, Nov 28 2005

Extended by Ray Chandler and Robert G. Wilson v, Nov 30 2005

Status

approved