A109291 New factors appearing in the factorization of 5^k - 2^k as k increases.

3, 7, 13, 29, 1031, 19, 25999, 641, 5563, 11, 41, 1409, 11551, 541, 406898311, 1597, 31, 8161, 17, 22993, 82009, 3101039, 37, 397, 6357828601279, 61, 5521, 43, 1009, 3613, 23, 303293, 7591, 197479, 2650751, 380881, 151, 95801, 6660751, 53, 131, 25117, 1271899175923

Offset: 1

Author

Jonathan Vos Post, Aug 25 2005

Keywords

• easy
• nonn

Comments

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.

Examples

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

Links

• Eric Weisstein's World of Mathematics, Zsigmondy's Theorem

Crossrefs

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

Programs

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

Extensions

Comment corrected by Jerry Metzger, Nov 04 2009

More terms from Michel Marcus, Nov 13 2016