login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A014945
Numbers k such that k divides 4^k - 1.
27
1, 3, 9, 21, 27, 63, 81, 147, 171, 189, 243, 441, 513, 567, 657, 729, 903, 1029, 1197, 1323, 1539, 1701, 1971, 2187, 2667, 2709, 3087, 3249, 3591, 3969, 4599, 4617, 5103, 5913, 6321, 6561, 7077, 7203, 8001, 8127, 8379, 9261, 9747, 10773, 11907, 12483
OFFSET
1,2
COMMENTS
This sequence is closed under multiplication. - Charles R Greathouse IV, Nov 03 2016
Conjecture: if k divides 4^k - 1, then (4^k - 1)/k is squarefree. - Thomas Ordowski, Dec 24 2018
Following Greathouse's comment, see A323203 for the primitive terms. - Bernard Schott, Jan 03 2019
All terms except 1 are divisible by 3. Proof: suppose n>1 is in the sequence, and let p be its smallest prime factor. Of course p is odd. Since 4^n-1 is divisible by p, n is divisible by the multiplicative order of 4 mod p, which is less than p. But since n has no prime factors < p, that multiplicative order can only be 1, which means p=3. - Robert Israel, Jan 24 2019
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..872 from Muniru A Asiru, terms 873..2000 from Alois P. Heinz)
FORMULA
a(n) = A014741(n+1)/2.
MAPLE
select(n->modp(4^n-1, n)=0, [$1..13000]); # Muniru A Asiru, Dec 28 2018
MATHEMATICA
Select[Range[12500], Divisible[4^#-1, #]&] (* Harvey P. Dale, Mar 23 2011 *)
PROG
(PARI) is(n)=Mod(4, n)^n==1 \\ Charles R Greathouse IV, Nov 03 2016
(GAP) a:=Filtered([1..13000], n->(4^n-1) mod n=0);; Print(a); # Muniru A Asiru, Dec 28 2018
(Magma) [n: n in [1..12500] | (4^n-1) mod n eq 0 ]; // Vincenzo Librandi, Dec 29 2018
(Python)
for n in range(1, 1000):
if (4**n-1) % n ==0:
print(n, end=', ') # Stefano Spezia, Jan 05 2019
CROSSREFS
Sequence in context: A044055 A029542 A014962 * A045590 A369768 A242740
KEYWORD
nonn
EXTENSIONS
More terms and better description from Benoit Cloitre, Mar 05 2002
STATUS
approved