A245973 -id:A245973 - cp's OEIS frontend

A245970 Tower of 2's modulo n.

0, 0, 1, 0, 1, 4, 2, 0, 7, 6, 9, 4, 3, 2, 1, 0, 1, 16, 5, 16, 16, 20, 6, 16, 11, 16, 7, 16, 25, 16, 2, 0, 31, 18, 16, 16, 9, 24, 16, 16, 18, 16, 4, 20, 16, 6, 17, 16, 23, 36, 1, 16, 28, 34, 31, 16, 43, 54, 48, 16, 22, 2, 16, 0, 16, 64, 17, 52, 52, 16, 3, 16

Offset: 1

Wayne VanWeerthuizen, Aug 08 2014

a(n) = (2^(2^(2^(2^(2^ ... ))))) mod n, provided enough 2's are in the tower so that adding more doesn't affect the value of a(n).

Let b(i) = A014221(i) = (2^(2^(2^(2^(2^ ... ))))), with i 2's. Since gcd(b(i)+1, b(j)+1) = gcd(2^2^b(i-2)+1, 2^2^b(j-2)+1) = gcd(A000215(b(i-2)), A000215(b(j-2))) = 1 for 1 <= i < j, there is no n > 1 such that a(n) = n-1. Since b(i)-1 = 2^2^b(i-2)-1 divides b(j)-1 = 2^2^b(j-2)-1 for 1 <= i < j, a(n) = 1 if and only if n > 1 is a divisor of a number of the form b(i)-1, or if and only if n > 1 is a divisor of a Fermat number (A023394). - Jianing Song, May 16 2024

			a(5) = 1, as 2^x mod 5 is 1 for x being any even multiple of two and X = 2^(2^(2^...)) is an even multiple of two.

Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.

Wayne VanWeerthuizen, Table of n, a(n) for n = 1..10000

Cf. A014221, A027763, A240162, A245971, A245972, A245973, A245974, A000010, A000079.

import Math.NumberTheory.Moduli (powerMod)
a245970 n = powerMod 2 (phi + a245970 phi) n
            where phi = a000010 n
-- Reinhard Zumkeller, Feb 01 2015

A:= proc(n)
     local phin,F,L,U;
     phin:= numtheory:-phi(n);
     if phin = 2^ilog2(phin) then
        F:= ifactors(n)[2];
        L:= map(t -> t[1]^t[2],F);
        U:= [seq(`if`(F[i][1]=2,0,1),i=1..nops(F))];
        chrem(U,L);
     else
        2 &^ A(phin) mod n
     fi
end proc:
seq(A(n), n=2 .. 100); # Robert Israel, Aug 19 2014

(* Import Mmca coding for "SuperPowerMod" and "LogStar" from text file in A133612 and then *) $RecursionLimit = 2^14; f[n_] := SuperPowerMod[2, 2^n, n] (* 2^^(2^n) (mod n), in Knuth's up-arrow notation *); Array[f, 72]
(* Second program: *)
a[n_] := Module[{phin, F, L, U},
   phin = EulerPhi[n];
   If[phin == 2^Floor@Log2[phin],
      F = FactorInteger[n];
      L = Power @@@ F;
      U = Table[If[F[[i, 1]] == 2, 0, 1], {i, 1, Length[F]}];
      ChineseRemainder[U, L],
      (2^a[phin])~Mod~n]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, May 03 2023, after Robert Israel *)

a(n)=if(n<3, return(0)); my(e=valuation(n,2),k=n>>e); lift(chinese(Mod(2,k)^a(eulerphi(k)), Mod(0,2^e))) \\ Charles R Greathouse IV, Jul 29 2016

def tower2mod(n):
    if ( n <= 22 ):
        return 65536%n
    else:
        ep = euler_phi(n)
        return power_mod(2,ep+tower2mod(ep),n)

a(n) = 2^(A000010(n)+a(A000010(n))) mod n.
a(n) = 0 if n is a power of 2.
a(n) = (2^2) mod n, if n < 5.
a(n) = (2^(2^2)) mod n, if n < 11.
a(n) = (2^(2^(2^2))) mod n, if n < 23.
a(n) = (2^(2^(2^(2^2)))) mod n, if n < 47.
a(n) = (2^^k) mod n, if n < A027763(k), where ^^ is Knuth's double-arrow notation.
From Robert Israel, Aug 19 2014: (Start)
If gcd(m,n) = 1, then a(m*n) is the unique k in [0,...,m*n-1] with
k == a(n) mod n and k == a(m) mod m.
a(n) = 1 if n is a Fermat number.
a(n) = 2^a(A000010(n)) mod n if n is not in A003401.
(End)

A240162 Tower of 3's modulo n.

Original entry on oeis.org

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

Wayne VanWeerthuizen, Aug 01 2014

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.

			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)

Wayne VanWeerthuizen, Table of n, a(n) for n = 1..10000

Cf. A245970, A245971, A245972, A245973, A245974.

Cf. A000010, A000244.

import Math.NumberTheory.Moduli (powerMod)
a245972 n = powerMod 3 (a245972 $ a000010 n) n
-- Reinhard Zumkeller, Feb 01 2015

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

a[1] = 0; a[n_] := a[n] = PowerMod[3, a[EulerPhi[n]], n]; Array[a, 72] (* Jean-François Alcover, Feb 09 2018 *)

def A(n):
    if ( n <= 10 ):
        return 27%n
    else:
        return power_mod(3,A(euler_phi(n)),n)

a(n) = 3^a(A000010(n)) mod n. - Robert Israel, Aug 01 2014

A245971 Tower of 4s mod n.

Original entry on oeis.org

0, 0, 1, 0, 1, 4, 4, 0, 4, 6, 4, 4, 9, 4, 1, 0, 1, 4, 9, 16, 4, 4, 3, 16, 21, 22, 13, 4, 24, 16, 4, 0, 4, 18, 11, 4, 34, 28, 22, 16, 37, 4, 41, 4, 31, 26, 17, 16, 11, 46, 1, 48, 47, 40, 26, 32, 28, 24, 45, 16, 57, 4, 4, 0, 61, 4, 55, 52, 49, 46, 50, 40, 37

Offset: 1

Wayne VanWeerthuizen, Aug 08 2014

a(n) = (4^(4^(4^(4^(4^ ... ))))) mod n, provided sufficient 4s are in the tower such that adding more doesn't affect the value of a(n).

Wayne VanWeerthuizen, Table of n, a(n) for n = 1..10000

Cf. A240162, A245970, A245972, A245973, A245974.

Cf. A000010, A000302.

import Math.NumberTheory.Moduli (powerMod)
a245971 n = powerMod 4 (phi + a245971 phi) n
            where phi = a000010 n
-- Reinhard Zumkeller, Feb 01 2015

def tower4mod(n):
    if n <= 10:
        return 256%n
    else:
        ep = euler_phi(n)
        return power_mod(4,ep+tower4mod(ep),n)
[tower4mod(n) for n in range(1, 30)]

A245972 Tower of 5s mod n.

Original entry on oeis.org

0, 1, 2, 1, 0, 5, 3, 5, 2, 5, 1, 5, 5, 3, 5, 5, 14, 11, 6, 5, 17, 1, 5, 5, 0, 5, 2, 17, 9, 5, 25, 21, 23, 31, 10, 29, 35, 25, 5, 5, 9, 17, 28, 1, 20, 5, 23, 5, 45, 25, 14, 5, 51, 29, 45, 45, 44, 9, 48, 5, 14, 25, 38, 53, 5, 23, 5, 65, 5, 45, 1, 29, 34, 35, 50

Offset: 1

Wayne VanWeerthuizen, Aug 08 2014

a(n) = (5^(5^(5^(5^(5^ ... ))))) mod n, provided sufficient 5s are in the tower such that adding more doesn't affect the value of a(n).

			a(2) = 1, as 5^X is odd for any whole number X.
a(19) = 6, as 5^(5^5) == 5^(5^(5^5)) == 5^(5^(5^(5^5))) == 6 (mod 19).

Wayne VanWeerthuizen, Table of n, a(n) for n = 1..10000

Cf. A000010, A240162, A245970, A245971, A245973, A245974.

A:= proc(n) option remember; 5 &^ A(numtheory:-phi(n)) mod n end proc:
A(2):= 1;
seq(A(n), n=2..100);

a[n_] := a[n] = PowerMod[5, If[n <= 18, 5, a[EulerPhi[n]]], n];
Array[a, 100] (* Jean-François Alcover, Jul 25 2022 *)

def a(n):
    if ( n <= 18 ):
        return 3125%n
    else:
        return power_mod(5,a(euler_phi(n)),n)

a(n) = 5^a(A000010(n)) mod n. For n<=18, a(n)=(5^5) mod n.

A245974 Tower of 7's mod n.

Original entry on oeis.org

0, 1, 1, 3, 3, 1, 0, 7, 7, 3, 2, 7, 6, 7, 13, 7, 12, 7, 7, 3, 7, 13, 20, 7, 18, 19, 16, 7, 1, 13, 19, 23, 13, 29, 28, 7, 34, 7, 19, 23, 26, 7, 7, 35, 43, 43, 37, 7, 0, 43, 46, 19, 11, 43, 13, 7, 7, 1, 7, 43, 6, 19, 7, 55, 58, 13, 63, 63, 43, 63, 66, 7, 30

Offset: 1

Wayne VanWeerthuizen, Aug 08 2014

a(n) = (7^(7^(7^(7^(7^ ... ))))) mod n, provided sufficient 7's are in the tower such that adding more doesn't affect the value of a(n).

			a(2) = 1, as 7^X is odd for any whole number X.
a(11) = 2, as 7^(7^7) == 7^(7^(7^7)) == 7^(7^(7^(7^7))) == 2 (mod 11).

Wayne VanWeerthuizen, Table of n, a(n) for n = 1..10000

Cf. A240162, A245970, A245971, A245972, A245973.

A:= proc(n) option remember; 7 &^ A(numtheory:-phi(n)) mod n end proc:
A(2):= 1;
seq(A(n), n=2..100);

a[n_] := a[n] = Switch[n, 1, 0, 2, 1, _, 7^a[EulerPhi[n]]]~Mod~n;
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Sep 21 2022 *)

def a(n):
    if ( n <= 10 ):
        return 823543%n
    else:
        return power_mod(7,a(euler_phi(n)),n)

a(n) = 7^a(A000010(n)) mod n. For n <= 10, a(n) = (7^7) mod n.

A332054 Tower of 9's modulo n.

Original entry on oeis.org

0, 1, 0, 1, 4, 3, 1, 1, 0, 9, 5, 9, 1, 1, 9, 9, 9, 9, 1, 9, 15, 5, 8, 9, 14, 1, 0, 1, 9, 9, 4, 9, 27, 9, 29, 9, 1, 1, 27, 9, 9, 15, 11, 5, 9, 31, 32, 9, 15, 39, 9, 1, 9, 27, 49, 1, 39, 9, 57, 9, 34, 35, 36, 9, 14, 27, 22, 9, 54, 29, 12, 9, 72, 1, 39, 1, 71, 27

Offset: 1

Jinyuan Wang, Mar 03 2020

nonn

a(n) = (9^(9^(9^(9^ ... )))) mod n, provided sufficient 9's are in the tower such that adding more doesn't affect the value of a(n).

Cf. A240162, A245970, A245971, A245972, A245973, A245974, A246497, A332055.

a(n) = {my(b, c=0, d=n, k=1, x=1); while(k==1, z=x; y=1; b=1; while(z>0, while(y

a(n) = 9^a(A000010(n)) mod n.

a(n) = (9^^k) mod n, if n < A246497(k), where ^^ is Knuth's double-arrow notation.

A332055 Tower of 8's modulo n.

Original entry on oeis.org

0, 0, 1, 0, 1, 4, 1, 0, 1, 6, 3, 4, 1, 8, 1, 0, 1, 10, 11, 16, 1, 14, 6, 16, 6, 14, 19, 8, 20, 16, 8, 0, 25, 18, 1, 28, 26, 30, 1, 16, 10, 22, 35, 36, 1, 6, 25, 16, 8, 6, 1, 40, 28, 46, 36, 8, 49, 20, 4, 16, 34, 8, 1, 0, 1, 58, 24, 52, 52, 36, 8, 64, 8, 26, 31

Offset: 1

Jinyuan Wang, Mar 04 2020

nonn

a(n) = (8^(8^(8^(8^ ... )))) mod n, provided sufficient 8's are in the tower such that adding more doesn't affect the value of a(n).

Cf. A240162, A245970, A245971, A245972, A245973, A245974, A246496, A332054.

a(n) = {my(b, c=0, d=n, k=1, x=1); while(k==1, z=x; y=1; b=1; while(z>0, while(y

a(n) = 8^(A000010(n) + a(A000010(n))) mod n.

a(n) = (8^^k) mod n, if n < A246496(k), where ^^ is Knuth's double-arrow notation.

A336111 A non-symmetrical rectangular array read by antidiagonals: A(n,m) is the tower of powers of n modulo m.

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 3, 1, 1, 0, 1, 4, 2, 0, 2, 0, 0, 1, 2, 3, 1, 1, 0, 1, 0, 1, 0, 6, 4, 0, 0, 1, 0, 0, 1, 7, 3, 4, 5, 1, 3, 1, 1, 0, 1, 6, 0, 0, 3, 0, 3, 0, 0, 0, 0, 1, 9, 7, 4, 5, 1, 1, 1, 1, 1, 1, 0, 1, 4, 9, 6, 2, 0, 0, 4, 4, 0, 2, 0, 0

Offset: 1

Jinyuan Wang and Robert G. Wilson v, Apr 15 2020

Although all numbers appear to be present, 1 appears most often followed by 0.

Since the first column and main diagonal are equal to 0, all matrices whose upper left corner is on the main diagonal have as their determinant 0.

			\m   1  2  3  4  5  6  7  8  9 10 11 12 13  14  15  16 ...
n\
_1   0  1  1  1  1  1  1  1  1  1  1  1  1   1   1   1
_2   0  0  1  0  1  4  2  0  7  6  9  4  3   2   1   0
_3   0  1  0  3  2  3  6  3  0  7  9  3  1  13  12  11
_4   0  0  1  0  1  4  4  0  4  6  4  4  9   4   1   0
_5   0  1  2  1  0  5  3  5  2  5  1  5  5   3   5   5
_6   0  0  0  0  1  0  1  0  0  6  5  0  1   8   6   0
_7   0  1  1  3  3  1  0  7  7  3  2  7  6   7  13   7
_8   0  0  1  0  1  4  1  0  1  6  3  4  1   8   1   0
_9   0  1  0  1  4  3  1  1  0  9  5  9  1   1   9   9
10   0  0  1  0  0  4  4  0  1  0  1  4  3   4  10   0
etc, .

Ilan Vardi, "Computational Recreations in Mathematica," Addison-Wesley Publishing Co., Redwood City, CA, 1991, pages 226-229.

Robert G. Wilson v, Mathematica coding for "SuperPowerMod" from Vardi

Cf. A245970, A240162, A245971, A245972, A245973, A245974, A332055, A332054.

(* first load all lines of Super Power Mod by Ilan Vardi from the hyper-link *)
Table[ SuperPowerMod[n - m + 1, 2^100, m], {n, 14}, {m, n, 1, -1}] // Flatten (* or *)
a[b_, 1] = 0; a[b_, n_] := PowerMod[b, If[OddQ@ b, a[b, EulerPhi[n]], EulerPhi[n] + a[b, EulerPhi[n]]], n]; Table[a[b - m + 1, m], {b, 14}, {m, b, 1, -1}] // Flatten

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A245970 Tower of 2's modulo n.

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A240162 Tower of 3's modulo n.

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A245971 Tower of 4s mod n.

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A245972 Tower of 5s mod n.

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A245974 Tower of 7's mod n.

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A332054 Tower of 9's modulo n.

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A332055 Tower of 8's modulo n.

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