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
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
Programs
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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
Comments