cp's OEIS Frontend

A109254 New factors appearing in the factorization of 7^k - 2^k as k increases.

%I A109254 #18 Jul 29 2025 00:21:48
%S A109254 5,3,67,53,11,61,13,164683,2417,163,739,1871,199,1987261,2221,1301,
%T A109254 14894543,71,1289,31,136261,17,339121,137,443,766606297,19,2017,
%U A109254 2279779036969771,5329741,43,235448977,23,9552313,47,116462754638606501,337,16993,101,158305897173001
%N A109254 New factors appearing in the factorization of 7^k - 2^k as k increases.
%C A109254 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 A109254 Harvey P. Dale, <a href="/A109254/b109254.txt">Table of n, a(n) for n = 1..249</a> (All terms through k = 100)
%H A109254 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ZsigmondyTheorem.html">Zsigmondy's Theorem</a>
%e A109254 a(1) = 5 because 7^1 - 2^1 = 5.
%e A109254 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 A109254 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 A109254 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 A109254 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 A109254 DeleteDuplicates[Flatten[FactorInteger[#][[All,1]]&/@Table[7^n-2^n,{n,50}]]] (* _Harvey P. Dale_, Apr 07 2022 *)
%o A109254 (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 A109254 Cf. A109325, A109347, A109348, A109349.
%K A109254 easy,nonn
%O A109254 1,1
%A A109254 _Jonathan Vos Post_, Aug 25 2005
%E A109254 Comment corrected by _Jerry Metzger_, Nov 04 2009
%E A109254 More terms from _Michel Marcus_, Nov 13 2016