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A276172 Number of primitive prime divisors of 3^n - 2^n.

%I A276172 #15 Feb 16 2025 08:33:36
%S A276172 0,1,1,1,1,1,2,1,1,1,2,1,2,1,1,2,1,1,2,1,2,1,2,1,2,2,1,1,1,2,1,2,2,2,
%T A276172 2,3,2,2,1,3,4,1,3,1,1,3,3,1,2,2,3,2,1,1,2,2,2,3,1,2,3,3,3,2,3,2,3,1,
%U A276172 5,1,2,3,3,2,3,1,3,2,4,1,3,4,3,2,2,5,3
%N A276172 Number of primitive prime divisors of 3^n - 2^n.
%C A276172 A prime factor of 3^n - 2^n is called primitive if it does not divide 3^r - 2^r for any positive r<n. In the general case with a^n -b^n for n>=2, Zsigmondy's theorem says that there is at least one primitive prime factor except two cases:
%C A276172 (i)   2^6 - 1^6
%C A276172 (ii)  n=2 and a+b is a power of 2.
%H A276172 Amiram Eldar, <a href="/A276172/b276172.txt">Table of n, a(n) for n = 1..568</a>
%H A276172 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ZsigmondyTheorem.html">Zsigmondy Theorem</a>.
%e A276172 a(7) = 2 because 3^7 - 2^7 = 2059 = 29*71 => 29 and 71 do not divide 3^r - 2^r  for r < 7:
%e A276172 3^1 - 2^1 = 1;
%e A276172 3^2 - 2^2 = 5;
%e A276172 3^3 - 2^3 = 19;
%e A276172 3^4 - 2^4 = 65 = 5*13;
%e A276172 3^5 - 2^5 = 211;
%e A276172 3^6 - 2^6 = 665 = 5*7*19.
%p A276172 f:= n -> nops(select(p -> numtheory:-order(3/2 mod p, p) = n, numtheory:-factorset(3^n-2^n)));
%p A276172 map(f, [$1..100]); # _Robert Israel_, Sep 14 2016
%t A276172 nMax=100; pLst={}; Table[f=Transpose[FactorInteger[3^n-2^n]][[1]]; f=Complement[f, pLst]; cnt=Length[f]; pLst=Union[pLst, f]; cnt, {n, 1, nMax}]
%Y A276172 Cf. A001047, A086251, A086257.
%K A276172 nonn
%O A276172 1,7
%A A276172 _Michel Lagneau_, Aug 23 2016
%E A276172 a(1) corrected by _Robert Israel_, Sep 14 2016