A245970 -id:A245970 - cp's OEIS frontend

A240162 Tower of 3's modulo n.

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.

A245973 Tower of 6s mod n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 0, 6, 5, 0, 1, 8, 6, 0, 1, 0, 1, 16, 15, 16, 2, 0, 6, 14, 0, 8, 23, 6, 1, 0, 27, 18, 1, 0, 1, 20, 27, 16, 18, 36, 1, 16, 36, 2, 36, 0, 43, 6, 18, 40, 47, 0, 16, 8, 39, 52, 9, 36, 9, 32, 36, 0, 1, 60, 14, 52, 48, 36, 6, 0, 1, 38, 6, 20

Offset: 1

Wayne VanWeerthuizen, Aug 08 2014

a(n) = (6^(6^(6^(6^(6^ ... ))))) mod n, provided sufficient 6s 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, A245971, A245972, A245974.

def tower6mod(n):
    if ( n <= 12 ):
        return 46656%n
    else:
        ep = euler_phi(n)
        return power_mod(6,ep+tower6mod(ep),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.

A256757 Number of iterations of A007733 required to reach 1.

Original entry on oeis.org

0, 1, 2, 1, 2, 2, 3, 1, 3, 2, 3, 2, 3, 3, 2, 1, 2, 3, 4, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 2, 3, 1, 3, 2, 3, 3, 4, 4, 3, 2, 3, 3, 4, 3, 3, 4, 5, 2, 4, 3, 2, 3, 4, 4, 3, 3, 4, 4, 5, 2, 3, 3, 3, 1, 3, 3, 4, 2, 4, 3, 4, 3, 4, 4, 3, 4, 3, 3, 4, 2, 5, 3, 4, 3, 2, 4, 4, 3, 4, 3, 3, 4, 3, 5, 4, 2, 3, 4, 3, 3

Offset: 1

Ivan Neretin, Apr 09 2015

In other words, the minimal height (not counting k) of the power tower 2^(2^(...^(2^k)...)) required to make it eventually constant modulo n (=A245970(n)) for sufficiently large k.

a(n) <= A227944(n) + 1. - Max Alekseyev, Oct 11 2016

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A007733, A256607 (second iteration), A256758 (positions of records), A003434, A227944 (similarly built upon the totient function).

a256757 n = fst $ until ((== 1) . snd)
            (\(i, x) -> (i + 1, fromIntegral $ a007733 x)) (0, n)
-- Reinhard Zumkeller, Apr 13 2015

A007733 = Function[n, MultiplicativeOrder[2, n/(2^IntegerExponent[n, 2])]];
a = Function[n, k = 0; m = n; While[m > 1, m = A007733[m]; k++]; k];
Table[a[n], {n, 100}] (* Ivan Neretin, Apr 13 2015 *)

a(n) = {if (n==1, return(0)); nb = 1; while((n = znorder(Mod(2, n/2^valuation(n, 2)))) != 1, nb++); nb;} \\ Michel Marcus, Apr 11 2015

For n>1, a(n) = a(A007733(n)) + 1.

A254411 Limit of f(f(f(...f(0)...))) modulo n as the number of iterations of f(x) = 2^x+1 grows.

Original entry on oeis.org

0, 1, 0, 1, 3, 3, 2, 1, 0, 3, 9, 9, 6, 9, 3, 1, 3, 9, 0, 13, 9, 9, 7, 9, 18, 19, 0, 9, 20, 3, 9, 1, 9, 3, 23, 9, 32, 19, 6, 33, 34, 9, 40, 9, 18, 7, 35, 33, 23, 43, 3, 45, 42, 27, 53, 9, 0, 49, 32, 33, 54, 9, 9, 1, 58, 9, 44, 37, 30, 23, 30, 9, 2, 69, 18, 57, 9, 45, 65, 33, 0, 75, 25, 9, 3, 83, 78, 9, 68, 63, 58, 53, 9, 35, 38, 33, 71, 23, 9, 93

Offset: 1

Max Alekseyev, Jan 30 2015

Also, limit of f(f(f(...f(m)...))) modulo n for any integer m >= 0.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A245970, A254410.

{ A254411(m) = my(g); if(m==1, return(0)); g=2^valuation(m,2); m\=g; lift( chinese(Mod(0,g),Mod(2,m)^A254411(eulerphi(m)) ) + 1) }

a(n) = limit of A254429(m) mod n as m grows.

a(n) = A254429(A227944(n)+k) mod n for any k>=1. In particular, a(n) = A254429(n) mod n.

A254410 Limit of f(f(f(...f(2)...))) modulo n as the number of iterations of f(x) = 2^x - 1 grows.

Original entry on oeis.org

0, 1, 1, 3, 2, 1, 1, 7, 1, 7, 6, 7, 10, 1, 7, 15, 8, 1, 1, 7, 1, 17, 17, 7, 2, 23, 1, 15, 26, 7, 3, 31, 28, 25, 22, 19, 34, 1, 10, 7, 4, 1, 1, 39, 37, 17, 35, 31, 1, 27, 25, 23, 32, 1, 17, 15, 1, 55, 36, 7, 5, 3, 1, 63, 62, 61, 43, 59, 40, 57, 49, 55, 1, 71, 52, 39, 50, 49, 75, 47, 1, 45, 66, 43, 42, 1, 55, 39, 63, 37, 36, 63, 34, 35, 77, 31, 65, 1, 28, 27

Offset: 1

Max Alekseyev, Jan 30 2015

Also, limit of f(f(f(...f(m)...))) modulo n for any integer m >= 2.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A245970, A254411.

Clear[a]; Unprotect[Power]; 0^0 = 1; a[1]=0; a[n_] := a[n] = Module[{g, m = n}, g = 2^IntegerExponent[m, 2]; m = Floor[m/g]; Mod[ ChineseRemainder[ {0, Mod[2, m]^a[EulerPhi[m]]}, {g, m}] - 1, n]]; Array[a, 100] (* Jean-François Alcover, Jan 01 2016, adapted from PARI *)

{ A254410(m) = my(g); if(m==1, return(0)); g=2^valuation(m,2); m\=g; lift( chinese(Mod(0,g),Mod(2,m)^A254410(eulerphi(m)) ) - 1) }

a(n) = limit of A007013(m) mod n as m grows.

a(n) = A007013(A227944(n) + k) mod n for any k >= 1. In particular, a(n) = A007013(n) mod n.

A318623 a(n) = 2^phi(n) mod n.

Original entry on oeis.org

0, 0, 1, 0, 1, 4, 1, 0, 1, 6, 1, 4, 1, 8, 1, 0, 1, 10, 1, 16, 1, 12, 1, 16, 1, 14, 1, 8, 1, 16, 1, 0, 1, 18, 1, 28, 1, 20, 1, 16, 1, 22, 1, 12, 1, 24, 1, 16, 1, 26, 1, 40, 1, 28, 1, 8, 1, 30, 1, 16, 1, 32, 1, 0, 1, 34, 1, 52, 1, 36, 1, 64, 1, 38, 1, 20, 1, 40, 1

Offset: 1

Jianing Song, Aug 30 2018

Of course, a(n) = 0 iff n is a power of 2 and a(n) = 1 iff n is an odd number > 1. For other n, let n = 2^t*s, t > 0, s > 1 is an odd number, then a(n) is the unique solution to x == 0 (mod 2^t) and x == 1 (mod s).

			a(6) = 2^phi(6) mod 6 = 2^4 mod 6 = 4.
a(18) = 2^phi(18) mod 18 = 2^6 mod 18 = 10.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000010, A007663, A245970.

[Modexp(2, EulerPhi(n), n): n in [1..110]]; // Vincenzo Librandi, Aug 02 2018

a[n_] = Mod[2^EulerPhi[n], n]; Array[a, 50] (* Stefano Spezia, Sep 01 2018 *)
Table[PowerMod[2,EulerPhi[n],n],{n,80}] (* Harvey P. Dale, Nov 07 2021 *)

PARI
```
a(n) = lift(Mod(2, n)^(eulerphi(n)))
```

If n is a power of 2 then a(n) = 0; if n is an odd number > 1 then a(n) = 1; else, let n = 2^t*s, t > 0, s > 1 is an odd number, then a(n) = n - (s mod 2^t)^2 + 1.

A318989 Limiting value of A318970(k) mod n as k grows.

Original entry on oeis.org

0, 1, 0, 1, 1, 3, 2, 5, 0, 1, 6, 9, 1, 9, 6, 5, 4, 9, 14, 1, 9, 17, 14, 21, 6, 1, 18, 9, 0, 21, 6, 5, 6, 21, 16, 9, 2, 33, 27, 21, 6, 9, 3, 17, 36, 37, 19, 21, 16, 31, 21, 1, 6, 45, 6, 37, 33, 29, 34, 21, 52, 37, 9, 5, 1, 39, 40, 21, 60, 51, 42, 45, 42, 39, 6, 33, 72, 27, 28, 21, 72, 47, 56, 9, 21, 3, 0, 61, 37, 81, 79, 37, 6, 19, 71, 69, 11, 65, 72, 81

Offset: 1

Max Alekseyev, Sep 06 2018

nonn

Is there a prime p in A318971 such that a(p) is nonzero?

Cf. A245970, A254410, A254411, A318970, A318971.

a(n) = A318970(k) mod n holds for any k >= A227944(n). In particular, a(n) = A318970(A227944(n)) mod n.

cp's OEIS Frontend

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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A245973 Tower of 6s mod n.

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A245974 Tower of 7's mod n.

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A256757 Number of iterations of A007733 required to reach 1.

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A254411 Limit of f(f(f(...f(0)...))) modulo n as the number of iterations of f(x) = 2^x+1 grows.

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