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A276172
Number of primitive prime divisors of 3^n - 2^n.
1
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, 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, 5, 1, 2, 3, 3, 2, 3, 1, 3, 2, 4, 1, 3, 4, 3, 2, 2, 5, 3
OFFSET
1,7
COMMENTS
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:
(i) 2^6 - 1^6
(ii) n=2 and a+b is a power of 2.
LINKS
Eric Weisstein's World of Mathematics, Zsigmondy Theorem.
EXAMPLE
a(7) = 2 because 3^7 - 2^7 = 2059 = 29*71 => 29 and 71 do not divide 3^r - 2^r for r < 7:
3^1 - 2^1 = 1;
3^2 - 2^2 = 5;
3^3 - 2^3 = 19;
3^4 - 2^4 = 65 = 5*13;
3^5 - 2^5 = 211;
3^6 - 2^6 = 665 = 5*7*19.
MAPLE
f:= n -> nops(select(p -> numtheory:-order(3/2 mod p, p) = n, numtheory:-factorset(3^n-2^n)));
map(f, [$1..100]); # Robert Israel, Sep 14 2016
MATHEMATICA
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Aug 23 2016
EXTENSIONS
a(1) corrected by Robert Israel, Sep 14 2016
STATUS
approved