A109325 Zsigmondy numbers for a = 3, b = 2: Zs(n, 3, 2) is the greatest divisor of 3^n - 2^n (A001047) that is relatively prime to 3^m - 2^m for all positive integers m < n.

1, 5, 19, 13, 211, 7, 2059, 97, 1009, 11, 175099, 61, 1586131, 463, 3571, 6817, 129009091, 577, 1161737179, 4621, 267331, 35839, 94134790219, 5521, 4015426801, 320503, 397760329, 369181, 68629840493971, 7471, 617671248800299, 43112257

Offset: 1

Author

Gottfried Helms, Aug 09 2005

Keywords

• nonn

Comments

The full factorization is multiplicative; meaning that the composition of factors is determined by the prime-factorization of n.

Examples

			Let n be 7; then the factorization of g(n) := 3^n-2^n is then g(7) = A(7) = 2059 since n is prime; let n be 3 then the factorization of g(3) = A(3) = 19 since n is prime; let n be 21, then the factorization is g(21) = A(3)*A(7)*A(21); and whether n is composite or not, with each n (at least) one new factor occurs besides the factors determined by the prime factors of n - so it is not purely multiplicative.
		

Links

• N. Bliss, B. Fulan, S. Lovett, and J. Sommars, Strong Divisibility, Cyclotomic Polynomials, and Iterated Polynomials, Amer. Math. Monthly, 120 (2013), 519-536.
• Eric Weisstein's World of Mathematics, Zsigmondy's Theorem

Crossrefs

Cf. A064078-A064083, A109347, A109348, A109349, A001047.

Programs

Maple

f:=proc(a,M) local n,b,d,t1,t2;
b:=[];
for n from 1 to M do
t1:=divisors(n);
t2:=mul(a[d]^mobius(n/d), d in t1);
b:=[op(b),t2];
od;
b;
end; a:=[seq(3^n-2^n,n=1..50)];
f(a,50); #  N. J. A. Sloane, Jun 07 2013

Formula

a(n) = Product_{d|n} b(d)^Moebius(n/d), where b() = A001047(). - N. J. A. Sloane, Jun 07 2013

Extensions

Edited and extended by Ray Chandler, Aug 26 2005