cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Wayne VanWeerthuizen

Wayne VanWeerthuizen's wiki page.

Wayne VanWeerthuizen has authored 17 sequences. Here are the ten most recent ones:

A246497 Smallest k such that 9^^n is not congruent to 9^^(n-1) mod k, where 9^^n denotes the power tower 9^9^...^9 (in which 9 appears n times).

Original entry on oeis.org

3, 7, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, 14619833, 36449279, 377982107, 1432349099
Offset: 1

Views

Author

Wayne VanWeerthuizen, Aug 27 2014

Keywords

Comments

For general remarks regarding sequences of this type, see A246491.
This particular sequence shares all known terms with A082449. they might be identical.

Crossrefs

Cf. A082449, A027763, A246491, A246492, A246493, A246494, A246495, A246496.

A246496 Smallest k such that 8^^n is not congruent to 8^^(n-1) mod k, where 8^^n denotes the power tower 8^8^...^8 (in which 8 appears n times).

Original entry on oeis.org

2, 3, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, 14920303
Offset: 1

Views

Author

Wayne VanWeerthuizen, Aug 27 2014

Keywords

Comments

For general remarks regarding sequences of this type, see A246491.

Crossrefs

Cf. A027763, A056637, A246491 - A246497
Large overlap with A056637.

A246495 Smallest k such that 7^^n is not congruent to 7^^(n-1) mod k, where 7^^n denotes the power tower 7^7^...^7 (in which 7 appears n times).

Original entry on oeis.org

4, 5, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, 4782969, 14348907
Offset: 1

Views

Author

Wayne VanWeerthuizen, Aug 27 2014

Keywords

Comments

For general remarks regarding sequences of this type, see A246491.

Crossrefs

Cf. A027763, A246491, A246493, A246494, A246496, A246497
Large overlap with A246492.

A246494 Smallest k such that 6^^n is not congruent to 6^^(n-1) mod k, where 6^^n denotes the power tower 6^6^...^6 (in which 6 appears n times).

Original entry on oeis.org

2, 4, 13, 47, 107, 643, 1439, 2879, 34549, 138197, 858239, 2029439, 36449279
Offset: 1

Views

Author

Wayne VanWeerthuizen, Aug 27 2014

Keywords

Comments

For general remarks regarding sequences of this type, see A246491.

Crossrefs

Cf. A027763, A246491, A246492, A246493, A246495, A246496, A246497

A246493 Smallest k such that 5^^n is not congruent to 5^^(n-1) mod k, where 5^^n denotes the power tower 5^5^...^5 (in which 5 appears n times).

Original entry on oeis.org

3, 7, 19, 47, 243, 719, 1439, 2879, 19683, 59049, 177147, 531441, 1594323, 4782969
Offset: 1

Views

Author

Wayne VanWeerthuizen, Aug 27 2014

Keywords

Comments

For general remarks regarding sequences of this type, see A246491.

Crossrefs

Cf. A027763, A246491, A246492, A246494, A246495, A246496, A246497

A246492 Smallest k such that 4^^n is not congruent to 4^^(n-1) mod k, where 4^^n denotes the power tower 4^4^...^4 (in which 4 appears n times).

Original entry on oeis.org

2, 5, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, 4782969, 14348907
Offset: 1

Views

Author

Wayne VanWeerthuizen, Aug 27 2014

Keywords

Comments

For general remarks regarding sequences of this type, see A246491.

Crossrefs

Cf. A027763, A246491, A246493, A246494, A246495, A246496, A246497

A246491 Smallest k such that 3^^n is not congruent to 3^^(n-1) mod k, where 3^^n denotes the power tower 3^3^...^3 (in which 3 appears n times).

Original entry on oeis.org

3, 5, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 1266767, 14920303, 36449279, 377982107
Offset: 1

Views

Author

Wayne VanWeerthuizen, Aug 27 2014

Keywords

Comments

This sequence is a member of an interesting class of sequences defined by the rule, "Smallest k such that b^^n is not congruent to b^^(n-1) mod k, where b^^n denotes the power tower b^b^...^b (in which b appears n times)," for some constant b. Different choices for b, with b>=2, determine a sequence of this class. The first sequence of this class is A027763, which uses b=2. This is the sequence for b=3. Adjacent sequences follow for b=4 through b=9.
Sequences of this class can contain only terms that are either prime numbers or powers of primes (A000961).
Powers of three (A000244) are commonplace as terms in sequences of this class, occurring significantly more often than powers of other primes.
For successive values of b, the sequence of first terms of all sequences of this class is A007978.
It appears that the sequence for b=x contains the term y, if and only if the sequence for b=x+A003418(y) also contains the term y, where A003418(y) is the least common multiple of all the integers from 1 to y. Can this be proved?
Sequences of this class generally share many terms with other sequences of this class, although each appears to be unique. These individual sequences are very similar to each other, so much so that among the first 7000 of them, only 120 distinct values less than 50000 occur (compare that to the 5217 available primes or powers of primes that are less than 50000.)
Sequences of this class tend to share many terms with A056637, the least prime of class n-.

Examples

			(3^^3) mod 11 = 9, (3^^2) mod 11 = 5. 11 is the least whole number for which this is true, thus a(3)=11.

Links

Crossrefs

Cf. A027763, A246492, A246493, A246494, A246495, A246496, A246497
Large overlap with A056637.

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

Views

Author

Wayne VanWeerthuizen, Aug 08 2014

Keywords

Comments

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).

Links

Crossrefs

Cf. A240162, A245970, A245971, A245972, A245974.

Programs

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

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

Views

Author

Wayne VanWeerthuizen, Aug 08 2014

Keywords

Comments

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).

Links

Crossrefs

Cf. A240162, A245970, A245972, A245973, A245974.
Cf. A000010, A000302.

Programs

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

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

Views

Author

Wayne VanWeerthuizen, Aug 01 2014

Keywords

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.

Examples

			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)

Links

Crossrefs

Cf. A245970, A245971, A245972, A245973, A245974.
Cf. A000010, A000244.

Programs

  • Haskell
    import Math.NumberTheory.Moduli (powerMod)
a245972 n = powerMod 3 (a245972 $ a000010 n) n
-- Reinhard Zumkeller, Feb 01 2015
  • 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 *)
  • Sage
    def A(n):
    if ( n <= 10 ):
        return 27%n
    else:
        return power_mod(3,A(euler_phi(n)),n)

Formula

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

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