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

2, 13, 5, 11, 7, 1093, 41, 757, 61, 23, 3851, 73, 797161, 547, 4561, 17, 193, 1871, 34511, 19, 37, 1597, 363889, 1181, 368089, 67, 661, 47, 1001523179, 6481, 8951, 391151, 398581, 109, 433, 8209, 29, 16493, 59, 28537, 20381027, 31, 271, 683

Offset: 1

Author

N. J. A. Sloane, May 13 2007

Keywords

• nonn

Comments

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

Except for k=1, there is at least one primitive prime divisor for every k. - T. D. Noe, Mar 01 2010

Links

• Max Alekseyev, Primes for k <= 690 (primes for k <= 500 from T. D. Noe)
• G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
• K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math., 3 (1892), 265-284.

Crossrefs

Cf. A003462, A076481, A028491.

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

Programs

Maple

# produce sequence
s1:=(a,b,M)->[seq( (a^n-b^n)/(a-b),n=0..M)];
# find primes and their indices
s2:=proc(s) local t1,t2,i; t1:=[]; t2:=[];
for i from 1 to nops(s) do if isprime(s[i]) then
t1:=[op(t1),s[i]];
t2:=[op(t2),i-1]; fi; od; RETURN(t1,t2); end;
# get primitive prime divisors in order
s3:=proc(s) local t2,t3,i,j,k,np; t2:=[]; np:=0;
for i from 1 to nops(s) do t3:=ifactors(s[i])[2];
for j from 1 to nops(t3) do p := t3[j][1]; new:=1;
for k from 1 to np do if p = t2[k] then new:= -1; break; fi; od;
if new = 1 then np:=np+1; t2:=[op(t2),p]; fi; od; od;
RETURN(t2); end;