A109291 - OEIS

%I #7 Nov 13 2016 12:38:16

%S 3,7,13,29,1031,19,25999,641,5563,11,41,1409,11551,541,406898311,1597,

%T 31,8161,17,22993,82009,3101039,37,397,6357828601279,61,5521,43,1009,

%U 3613,23,303293,7591,197479,2650751,380881,151,95801,6660751,53,131,25117,1271899175923

%N New factors appearing in the factorization of 5^k - 2^k as k increases.

%C Zsigmondy numbers for a = 5, b = 2: Zs(n, 5, 2) is the greatest divisor of 5^k - 2^k that is relatively prime to 5^j - 2^j for all positive integers j < k.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ZsigmondyTheorem.html">Zsigmondy's Theorem</a>

%e a(1) = 3 because 5^1 - 2^1 = 3.

%e a(2) = 7 because, although 5^2 - 2^2 = 21 = 3 * 7 has prime factor 3, that has already appeared in this sequence, but the factor of 7 is new.

%e a(3) = 13 because, although 5^3 - 2^3 = 117 = 3^2 * 13 has repeated prime factor 3, that has already appeared in this sequence, but the prime factor of 13 is new.

%e a(4) = 29 because, although 5^4 - 2^4 = 2385 = 609 = 3 * 7 * 29, the prime factors of 3 and 7 have already appeared in this sequence, but the prime factor of 29 is new.

%e a(5) = 1031 because, although 5^5 - 2^5 = 16775 = 3093 = 3 * 1031, the prime factor of 3 has already appeared in this sequence, but the prime factors of 1031 is new.

%o (PARI) lista(nn) = {my(pf = []); for (k=1, nn, f = factor(5^k-2^k)[,1]; for (j=1, #f~, if (!vecsearch(pf, f[j]), print1(f[j], ", "); pf = vecsort(concat(pf, f[j])));););} \\ _Michel Marcus_, Nov 13 2016

%Y Cf. A109325, A109347, A109348, A109349, A109254.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Aug 25 2005

%E Comment corrected by _Jerry Metzger_, Nov 04 2009

%E More terms from _Michel Marcus_, Nov 13 2016