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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. 7

%I #26 May 26 2019 16:07:54

%S 2,3,5,11,7,23,13,29,41,17,53,37,83,43,89,19,113,131,67,47,73,31,79,

%T 173,179,61,191,97,233,239,251,127,139,281,71,293,101,103,107,163,59,

%U 359,193,199,137,419,431,443,151,491,509,181,109,277,593,149,307,641,653

%N 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.

%C 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

%C If "p-1" is changed to "p+1" we get A236388. - _N. J. A. Sloane_, Jan 24 2014

%C 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

%H Alois P. Heinz, <a href="/A112037/b112037.txt">Table of n, a(n) for n = 2..10000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions">Dirichlet's theorem on arithmetic progressions</a>

%e We start with the second prime, 3. 3-1 = 2, so 2 is the first term.

%e 5-1 = 2*2, nothing new.

%e 7-1 = 2*3 and 3 is new, so that is the second term.

%e 11-1 = 2*5 and we get 5; etc.

%t 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 *)

%t DeleteDuplicates[Rest[Flatten[FactorInteger[#][[All,1]]&/@ (Prime[ Range[ 250]]-1)]]] (* _Harvey P. Dale_, May 26 2019 *)

%o (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

%o (GAP) Set(Flat(List(Filtered([3..1500],IsPrime),i->Factors(i-1)))); # _Muniru A Asiru_, Dec 06 2018

%Y Cf. A055768, A112038, A114461, A236388.

%K easy,nonn

%O 2,1

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

%E Better description from _Jack Brennen_, Nov 28 2005

%E Extended by _Ray Chandler_ and _Robert G. Wilson v_, Nov 30 2005

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