1,1

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. We show only through k = 20, ready for extension and Mathematica.

MathWorld, Zsigmondy's Theorem

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.

Cf. A109325, A109347, A109348, A109349, my submission of 10 minutes ago Zs(n, 7, 2).

Sequence in context: A091565 A025249 A147098 this_sequence A062700 A136060 A023227

Adjacent sequences: A109288 A109289 A109290 this_sequence A109292 A109293 A109294

easy,nonn,new

Jonathan Vos Post (jvospost3(AT)gmail.com), Aug 25 2005

Comment corrected by Jerry Metzger, Nov 04 2009