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

1,1

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.

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

Cf. A109325, A109347, A109348, A109349.

Sequence in context: A181755 A007299 A257935 * A258091 A370551 A350213

Adjacent sequences: A109251 A109252 A109253 * A109255 A109256 A109257

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