login
A109254
New factors appearing in the factorization of 7^k - 2^k as k increases.
2
5, 3, 67, 53, 11, 61, 13, 164683, 2417, 163, 739, 1871, 199, 1987261, 2221, 1301, 14894543, 71, 1289, 31, 136261, 17, 339121, 137, 443, 766606297, 19, 2017, 2279779036969771, 5329741, 43, 235448977, 23, 9552313, 47, 116462754638606501, 337, 16993, 101, 158305897173001
OFFSET
1,1
COMMENTS
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.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..249 (* All terms through k = 100 *)
Eric Weisstein's World of Mathematics, Zsigmondy's Theorem
EXAMPLE
a(1) = 5 because 7^1 - 2^1 = 5.
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.
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.
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.
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.
MATHEMATICA
DeleteDuplicates[Flatten[FactorInteger[#][[All, 1]]&/@Table[7^n-2^n, {n, 50}]]] (* Harvey P. Dale, Apr 07 2022 *)
PROG
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Aug 25 2005
EXTENSIONS
Comment corrected by Jerry Metzger, Nov 04 2009
More terms from Michel Marcus, Nov 13 2016
STATUS
approved