A109254 - OEIS

%I #13 Apr 07 2022 10:46:02

%S 5,3,67,53,11,61,13,164683,2417,163,739,1871,199,1987261,2221,1301,

%T 14894543,71,1289,31,136261,17,339121,137,443,766606297,19,2017,

%U 2279779036969771,5329741,43,235448977,23,9552313,47,116462754638606501,337,16993,101,158305897173001

%N New factors appearing in the factorization of 7^k - 2^k as k increases.

%C Zsigmondy numbers for a = 7, b = 2: Zs(n, 7, 2) is the greatest divisor of 7^k - 2^k that is relatively prime to 7^j - 2^j for all positive integers j < k.

%H Harvey P. Dale, <a href="/A109254/b109254.txt">Table of n, a(n) for n = 1..249</a> (* All terms through k = 100 *)

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

%e a(1) = 5 because 7^1 - 2^1 = 5.

%e a(2) = 3 because, although 7^2 - 2^2 = 45 = 3^2 * 5 has prime factor 5, that has already appeared in this sequence, but the repeated prime factor of 3 is new.

%e a(3) = 67 because, although 7^3 - 2^3 = 335 = 5 * 67 has prime factor 5, that has already appeared in this sequence, but the prime factor of 67 is new.

%e a(4) = 53 because, although 7^4 - 2^4 = 2385 = 3^2 * 5 * 53, the prime factors of 3 and 5 have already appeared in this sequence, but the prime factor of 53 is new.

%e a(5) = 11 and a(6) = 61 because, although 7^5 - 2^5 = 16775 = 5^2 * 11 * 61, the prime factor of 5 has already appeared in this sequence, but the prime factors of 11 and 61 are new.

%t DeleteDuplicates[Flatten[FactorInteger[#][[All,1]]&/@Table[7^n-2^n,{n,50}]]] (* _Harvey P. Dale_, Apr 07 2022 *)

%o (PARI) lista(nn) = {my(pf = []); for (k=1, nn, f = factor(7^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.

%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