|
|
A109254
|
|
New factors appearing in the factorization of 7^k - 2^k as k increases.
|
|
2
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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 A350213 A255599
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
|
|
|
|