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A129733 List of primitive prime divisors of the numbers (3^k-1)/2 (A003462) for k>=2, in order of their occurrence.

%I A129733 #23 Dec 14 2024 07:18:46
%S A129733 2,13,5,11,7,1093,41,757,61,23,3851,73,797161,547,4561,17,193,1871,
%T A129733 34511,19,37,1597,363889,1181,368089,67,661,47,1001523179,6481,8951,
%U A129733 391151,398581,109,433,8209,29,16493,59,28537,20381027,31,271,683
%N A129733 List of primitive prime divisors of the numbers (3^k-1)/2 (A003462) for k>=2, in order of their occurrence.
%C A129733 Read A003462 term-by-term, factorize each term, write down any primes not seen before.
%C A129733 Except for k=1, there is at least one primitive prime divisor for every k. - _T. D. Noe_, Mar 01 2010
%H A129733 Max Alekseyev, <a href="/A129733/b129733.txt">Primes for k <= 690</a> (primes for k <= 500 from T. D. Noe)
%H A129733 G. Everest et al., <a href="http://www.jstor.org/stable/27642221">Primes generated by recurrence sequences</a>, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
%H A129733 K. Zsigmondy, <a href="https://doi.org/10.1007/BF01692444">Zur Theorie der Potenzreste</a>, Monatsh. Math., 3 (1892), 265-284.
%p A129733 # produce sequence
%p A129733 s1:=(a,b,M)->[seq( (a^n-b^n)/(a-b),n=0..M)];
%p A129733 # find primes and their indices
%p A129733 s2:=proc(s) local t1,t2,i; t1:=[]; t2:=[];
%p A129733 for i from 1 to nops(s) do if isprime(s[i]) then
%p A129733 t1:=[op(t1),s[i]];
%p A129733 t2:=[op(t2),i-1]; fi; od; RETURN(t1,t2); end;
%p A129733 # get primitive prime divisors in order
%p A129733 s3:=proc(s) local t2,t3,i,j,k,np; t2:=[]; np:=0;
%p A129733 for i from 1 to nops(s) do t3:=ifactors(s[i])[2];
%p A129733 for j from 1 to nops(t3) do p := t3[j][1]; new:=1;
%p A129733 for k from 1 to np do if p = t2[k] then new:= -1; break; fi; od;
%p A129733 if new = 1 then np:=np+1; t2:=[op(t2),p]; fi; od; od;
%p A129733 RETURN(t2); end;
%Y A129733 Cf. A003462, A076481, A028491.
%Y A129733 If 3 is replaced with 2, we get A000225, A000043, A108974 respectively.
%K A129733 nonn
%O A129733 1,1
%A A129733 _N. J. A. Sloane_, May 13 2007