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A014945
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Numbers k such that k divides 4^k - 1.
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27
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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
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OFFSET
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1,2
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COMMENTS
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Conjecture: if k divides 4^k - 1, then (4^k - 1)/k is squarefree. - Thomas Ordowski, Dec 24 2018
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
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LINKS
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FORMULA
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MAPLE
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select(n->modp(4^n-1, n)=0, [$1..13000]); # Muniru A Asiru, Dec 28 2018
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MATHEMATICA
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Select[Range[12500], Divisible[4^#-1, #]&] (* Harvey P. Dale, Mar 23 2011 *)
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PROG
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(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:
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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