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

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

Author

Jonathan Vos Post, Aug 25 2005

Keywords

• easy
• nonn

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.

Examples

			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.
		

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

Crossrefs

Cf. A109325, A109347, A109348, A109349.

Programs

Mathematica

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

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

Extensions

Comment corrected by Jerry Metzger, Nov 04 2009

More terms from Michel Marcus, Nov 13 2016