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

1,1

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.

Links

Table of n, a(n) for n=1..43.

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

Example

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.

Prog

(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

Crossrefs

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

Sequence in context: A091565 A025249 A147098 * A340067 A199218 A062700

Adjacent sequences: A109288 A109289 A109290 * A109292 A109293 A109294

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