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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.

%I A109325 #18 Feb 16 2025 08:32:58
%S A109325 1,5,19,13,211,7,2059,97,1009,11,175099,61,1586131,463,3571,6817,
%T A109325 129009091,577,1161737179,4621,267331,35839,94134790219,5521,
%U A109325 4015426801,320503,397760329,369181,68629840493971,7471,617671248800299,43112257
%N 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.
%C A109325 The full factorization is multiplicative; meaning that the composition of factors is determined by the prime-factorization of n.
%H A109325 N. Bliss, B. Fulan, S. Lovett, and J. Sommars, <a href="http://dx.doi.org/10.4169/amer.math.monthly.120.06.519">Strong Divisibility, Cyclotomic Polynomials, and Iterated Polynomials</a>, Amer. Math. Monthly, 120 (2013), 519-536.
%H A109325 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ZsigmondyTheorem.html">Zsigmondy's Theorem</a>
%F A109325 a(n) = Product_{d|n} b(d)^Moebius(n/d), where b() = A001047(). - _N. J. A. Sloane_, Jun 07 2013
%e A109325 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.
%p A109325 f:=proc(a,M) local n,b,d,t1,t2;
%p A109325 b:=[];
%p A109325 for n from 1 to M do
%p A109325 t1:=divisors(n);
%p A109325 t2:=mul(a[d]^mobius(n/d), d in t1);
%p A109325 b:=[op(b),t2];
%p A109325 od;
%p A109325 b;
%p A109325 end; a:=[seq(3^n-2^n,n=1..50)];
%p A109325 f(a,50); #  _N. J. A. Sloane_, Jun 07 2013
%Y A109325 Cf. A064078-A064083, A109347, A109348, A109349, A001047.
%K A109325 nonn
%O A109325 1,2
%A A109325 _Gottfried Helms_, Aug 09 2005
%E A109325 Edited and extended by _Ray Chandler_, Aug 26 2005