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A240162
Tower of 3's modulo n.
10
0, 1, 0, 3, 2, 3, 6, 3, 0, 7, 9, 3, 1, 13, 12, 11, 7, 9, 18, 7, 6, 9, 18, 3, 12, 1, 0, 27, 10, 27, 23, 27, 9, 7, 27, 27, 36, 37, 27, 27, 27, 27, 2, 31, 27, 41, 6, 27, 6, 37, 24, 27, 50, 27, 42, 27, 18, 39, 49, 27, 52, 23, 27, 59, 27, 9, 52, 7, 18, 27, 49, 27
OFFSET
1,4
COMMENTS
a(n) = (3^(3^(3^(3^(3^ ... ))))) mod n, provided sufficient 3's are in the tower such that adding more doesn't affect the value of a(n).
For values of n significantly less than Graham's Number, a(n) is equal to Graham's Number mod n.
LINKS
Wayne VanWeerthuizen, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = 3^a(A000010(n)) mod n. - Robert Israel, Aug 01 2014
EXAMPLE
a(7) = 6. For any natural number X, 3^X is a positive odd multiple of 3. 3^(any positive odd multiple of three) mod 7 is always 6.
a(9) = 0, since 3^(3^X) is divisible by 9 for any natural number X. In our case, X itself is a tower of 3's.
a(100000000) = 64195387, giving the rightmost eight digits of Graham's Number.
From Robert Munafo, Apr 19 2020: (Start)
a(1) = 0, because 3 mod 1 = 0.
a(2) = 1, because 3^3 mod 2 = 1.
a(3) = 0, because 3^3^3 mod 3 = 0.
a(4) = 3, because 3^3^3^3 = 3^N for odd N, 3^N = 3 mod 4 for all odd N.
a(5) = 3^3^3^3^3 mod 5, and we should look at the sequence 3^N mod 5. We find that 3^N = 2 mod 5 whenever N = 3 mod 4. As just shown in the a(4) example, 3^3^3^3 = 3 mod 4. (End)
MAPLE
A:= proc(n) option remember; 3 &^ A(numtheory:-phi(n)) mod n end proc:
A(2):= 1;
seq(A(n), n=2..100); # Robert Israel, Aug 01 2014
MATHEMATICA
a[1] = 0; a[n_] := a[n] = PowerMod[3, a[EulerPhi[n]], n]; Array[a, 72] (* Jean-François Alcover, Feb 09 2018 *)
PROG
(Sage)
def A(n):
if ( n <= 10 ):
return 27%n
else:
return power_mod(3, A(euler_phi(n)), n)
(Haskell)
import Math.NumberTheory.Moduli (powerMod)
a245972 n = powerMod 3 (a245972 $ a000010 n) n
-- Reinhard Zumkeller, Feb 01 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved