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List of primitive prime divisors of the numbers (3^k-1)/2 (A003462) for k>=2, in order of their occurrence.
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%I #23 Dec 14 2024 07:18:46

%S 2,13,5,11,7,1093,41,757,61,23,3851,73,797161,547,4561,17,193,1871,

%T 34511,19,37,1597,363889,1181,368089,67,661,47,1001523179,6481,8951,

%U 391151,398581,109,433,8209,29,16493,59,28537,20381027,31,271,683

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

%C Read A003462 term-by-term, factorize each term, write down any primes not seen before.

%C Except for k=1, there is at least one primitive prime divisor for every k. - _T. D. Noe_, Mar 01 2010

%H Max Alekseyev, <a href="/A129733/b129733.txt">Primes for k <= 690</a> (primes for k <= 500 from T. D. Noe)

%H 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 K. Zsigmondy, <a href="https://doi.org/10.1007/BF01692444">Zur Theorie der Potenzreste</a>, Monatsh. Math., 3 (1892), 265-284.

%p # produce sequence

%p s1:=(a,b,M)->[seq( (a^n-b^n)/(a-b),n=0..M)];

%p # find primes and their indices

%p s2:=proc(s) local t1,t2,i; t1:=[]; t2:=[];

%p for i from 1 to nops(s) do if isprime(s[i]) then

%p t1:=[op(t1),s[i]];

%p t2:=[op(t2),i-1]; fi; od; RETURN(t1,t2); end;

%p # get primitive prime divisors in order

%p s3:=proc(s) local t2,t3,i,j,k,np; t2:=[]; np:=0;

%p for i from 1 to nops(s) do t3:=ifactors(s[i])[2];

%p for j from 1 to nops(t3) do p := t3[j][1]; new:=1;

%p for k from 1 to np do if p = t2[k] then new:= -1; break; fi; od;

%p if new = 1 then np:=np+1; t2:=[op(t2),p]; fi; od; od;

%p RETURN(t2); end;

%Y Cf. A003462, A076481, A028491.

%Y If 3 is replaced with 2, we get A000225, A000043, A108974 respectively.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, May 13 2007