A246491 -id:A246491 - cp's OEIS frontend

A027763 Smallest k such that 2^^n is not congruent to 2^^(n-1) mod k, where 2^^n denotes the power tower 2^2^...^2 (in which 2 appears n times).

2, 3, 5, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 531441, 1594323, 4782969, 14348907, 43046721, 86093443, 344373773, 688747547, 3486784401

Offset: 1

R. K. Guy

This sequence shares many terms with A056637, the least prime of class n-. Note that 3^(n-1) is an upper bound for each term and the upper bound is reached for n=13 and n=14. Are all subsequent terms 3^(n-1)? The Mathematica code uses the TowerMod function in the CNT package, which is described in the book by Bressoud and Wagon. - T. D. Noe, Mar 13 2009
For n=15, n=16, and n=17, the terms are also of the form 3^(n-1), but for n=18 and n=19, the terms are prime. - Wayne VanWeerthuizen, Aug 26 2014
A185816(a(n)) = n. - Reinhard Zumkeller, Sep 02 2014
Prime terms seen up to n=20 are in eleven instances of the form j*a(n-1)+1, for j=2, 4, 6, or 12. Note, though, that a(2)=5 and a(8)=719 are exceptions to this pattern. - Wayne VanWeerthuizen, Sep 06 2014

			2^^2=2^2=4 and 2^^3=2^2^2=16. We find 4 = 16 (mod k) until k=5. So a(3)=5. - _T. D. Noe_, Mar 13 2009

David Bressoud and Stan Wagon, A Course in Computational Number Theory, Key College Pub., 2000, p. 96.
Stan Wagon, posting to Problem of the Week mailing list, Dec 15 1997.

D. Bressoud, CNT.m Computational Number Theory Mathematica package.
Stan Wagon, Putnam Problem Notes
Eric W. Weisstein, MathWorld: Power Tower

Cf. A056637, A173927, A185816, A246491.

Needs["CNT`"]; k=1; Table[While[TowerMod[2,n,k]==TowerMod[2,n-1,k], k++ ]; k, {n,10}] (* T. D. Noe, Mar 13 2009 *)

Improved the name and changed the offset because I just prepended a term. - T. D. Noe, Mar 13 2009
Corrected and extended by T. D. Noe, Mar 13 2009
Terms a(15)-a(19) from Wayne VanWeerthuizen, Aug 26 2014
Terms a(20)-a(21) from Wayne VanWeerthuizen, Sep 06 2014

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

Wayne VanWeerthuizen, Aug 27 2014

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

Cf. A027763, A056637, A246491 - A246497

Large overlap with A056637.

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

Wayne VanWeerthuizen, Aug 27 2014

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

This particular sequence shares all known terms with A082449. they might be identical.

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

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

Wayne VanWeerthuizen, Aug 27 2014

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

Cf. A027763, A246491, A246493, A246494, 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

Wayne VanWeerthuizen, Aug 27 2014

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

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

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

Wayne VanWeerthuizen, Aug 27 2014

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

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

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

Wayne VanWeerthuizen, Aug 27 2014

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

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

Large overlap with A246492.

A379259 a(n) is the number of iterations that n requires to reach a 3-smooth number under the map x -> phi(x).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 19 2024

If k is a 3-smooth number then phi(k) is also a 3-smooth number. Therefore, a(n) counts the numbers that are not 3-smooth numbers in the trajectory from n to a 3-smooth number (including n if it is not a 3-smooth number).

The indices of records, 1, 5, 11, 23, 47, ..., seem to be A246491 except for the first term (checked up to A246491(15)).

			a(1) = a(2) = a(3) = a(4) = 0 because 1, 2, 3 and 4 are already 3-smooth numbers.
a(5) = 1 because after one iteration 5 -> phi(5) = 4, a 3-smooth number, 4, is reached.
a(23) = 3 because after 3 iterations 23 -> 22 -> 10 -> 4, a 3-smooth number, 4, is reached.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A003434, A003586, A086420, A122254, A246491.

smQ[n_] := n == Times @@ ({2, 3}^IntegerExponent[n, {2, 3}]); a[n_] := -1 + Length@ NestWhileList[EulerPhi, n, ! smQ[#] &]; Array[a, 100]

issm(n) = my(m = n >> valuation(n, 2)); m == 3^valuation(m, 3);
a(n) = {my(c = 0); while(!issm(n), c++; n = eulerphi(n)); c;}

a(A003586(n)) = 0.

cp's OEIS Frontend

A027763 Smallest k such that 2^^n is not congruent to 2^^(n-1) mod k, where 2^^n denotes the power tower 2^2^...^2 (in which 2 appears n times).

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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).

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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).

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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).

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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).

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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).

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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).

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A379259 a(n) is the number of iterations that n requires to reach a 3-smooth number under the map x -> phi(x).

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Mathematica

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