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

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

Amiram Eldar, Table of n, a(n) for n = 1..568

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

Cf. A001047, A086251, A086257.

Sequence in context: A218545 A205600 A133162 * A322028 A079806 A342845

Adjacent sequences: A276169 A276170 A276171 * A276173 A276174 A276175

Keyword

nonn

Author

Michel Lagneau, Aug 23 2016

Extensions

a(1) corrected by Robert Israel, Sep 14 2016

Status

approved