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A268081
Least positive integer k such that 3^n-1 and k^n-1 are relatively prime.
2
2, 2, 2, 10, 2, 28, 2, 10, 2, 22, 10, 910, 2, 2, 2, 170, 2, 3458, 2, 110, 2, 46, 10, 910, 2, 2, 2, 290, 2, 9548, 2, 340, 10, 2, 22, 639730, 2, 2, 2, 4510, 2, 1204, 10, 230, 2, 94, 2, 216580, 2, 22, 2, 530, 2, 3458, 22, 580, 2, 118, 2, 18928910
OFFSET
1,1
COMMENTS
Note that (3^n-1)^n-1 is always relatively prime to 3^n-1.
According to the conjecture given in A086892, a(n) = 2 infinitely often.
When n>1, a(n) = 2 if and only if A260119(n) = 3.
From Robert Israel, Nov 20 2024: (Start)
a(n) <= a(m*n) for m >= 1.
If p is a prime factor of 3^n - 1 such that p-1 divides n, then a(n) is a multiple of p. (End)
LINKS
EXAMPLE
Since 3^5-1 = 242 and 2^5-1 = 31 are relatively prime, a(5) = 2.
MAPLE
f:= proc(n) local t, F, m, k, v;
t:= 3^n-1;
F:= select(isprime, map(`+`, numtheory:-divisors(n), 1));
m:= convert(select(s -> t mod s = 0, F), `*`);
for k from m by m do
v:= k &^ n - 1 mod t;
if igcd(v, t) = 1 then return k fi
od
end proc:
map(f, [$1..100]); # Robert Israel, Nov 20 2024
MATHEMATICA
Table[k = 1; While[! CoprimeQ[3^n - 1, k^n - 1], k++]; k, {n, 59}] (* Michael De Vlieger, Jan 27 2016 *)
PROG
(Sage)
def min_k(n):
g, k=2, 0
while g!=1:
k=k+1
g=gcd(3^n-1, k^n-1)
return k
print([min_k(n) for n in [1..60]])
(PARI) a(n) = {k=1; while( gcd(3^n-1, k^n-1)!=1, k++); k; }
CROSSREFS
Sequence in context: A091185 A372723 A324956 * A319885 A368958 A125695
KEYWORD
nonn,changed
AUTHOR
Tom Edgar, Jan 25 2016
STATUS
approved