A055777 -id:A055777 - cp's OEIS frontend

A364855 Initial digit of 3^(3^n) (A055777(n)).

3, 2, 1, 7, 4, 8, 6, 2, 2, 1, 3, 3, 6, 2, 1, 3, 3, 4, 6, 2, 2, 1, 1, 1, 5, 1, 2, 1, 1, 7, 4, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 7, 4, 8, 6, 2, 1, 2, 1, 3, 4, 1, 1, 1, 4, 8, 6, 2, 2, 1, 2, 2, 1, 5, 1, 6, 3, 3, 4, 1, 1, 2, 1, 5, 1, 4, 1

Offset: 0

Marco Ripà, Aug 10 2023

This sequence corresponds to the initial digit of 3vvn (since 3^(3^n) = ((((3^3)^3)^...)^3) n-times), where vv indicates weak tetration (see links).
The author conjectures that the distribution of the initial digits of the present sequence obey Benford's law or Zipf's law (see links).
The corresponding final digit of 3^(3^n) is A010705(n) = 3 if n even or 7 if n odd.

			a(2) = 1, since 3^(3^2) = 3^9 = 19683.

A. Iorliam, Natural Laws (Benford's Law and Zipf's Law) For Network Traffic Analysis, In: Cybersecurity in Nigeria. SpringerBriefs in Cybersecurity. Springer, Cham (2019), 3-22. DOI: 10.1007/978-3-030-15210-9_2

Pointless Large numbers stuff by Cookiefonster, 2.03 The Weak Hyper-Operators.
Wikipedia, Benford's law.
Wikipedia, Zipf's law.

Cf. A000030, A010705 (last digit), A055777, A364789, A364837.

Join[{3},Table[Floor[3^(3^n)/10^Floor[Log10[3^(3^n)]]],{n,16}]]

a(n) = floor(3^(3^n)/10^floor(log_10(3^(3^n)))).

a(n) = A000030(A055777(n)).

More terms from Jinyuan Wang, Aug 11 2023

A129290 Base-3 Fermat numbers: 3^(3^n) + 1.

Original entry on oeis.org

4, 28, 19684, 7625597484988, 443426488243037769948249630619149892804, 87189642485960958202911070585860771696964072404731750085525219437990967093723439943475549906831683116791055225665628

Offset: 0

Alexander Adamchuk, Apr 08 2007

nonn

Apparently discovered (with arbitrary base) by Gottschalk in 1938 and independently by Ferentinou-Nicolacopoulou in 1963. - Charles R Greathouse IV, Jul 05 2011

a(n) divides a(n+1). a(n+1)/a(n) = (3^(3^(n+1)) + 1)/(3^(3^n) + 1) = 1 - 3^(3^n) + 9^(3^n) = A002061(3^(3^n)) = A129291(n) = {7, 703, 387400807, 58149737003032434092905183, ...}.

J. Ferentinou-Nicolacopoulou, "Une propriété des diviseurs du nombre r^(r^m)+1. Applications au dernier théorème de Fermat." Bulletin Société Mathématique de Grèce 4:1 (1963), pp. 121-126.

Eugen Gottschalk, Zum Fermatschen Problem, Mathematische Annalen 115 (1934), pp. 157-158.
Lorenzo Sauras-Altuzarra, Some properties of the factors of Fermat numbers, Art Discrete Appl. Math. (2022).

Cf. A000215 (Fermat numbers: 2^(2^n) + 1).
Cf. A055777 (3^(3^n)).
Cf. A129291 (A129290(n+1) / A129290(n)).
Cf. A002061 (central polygonal numbers: n^2 - n + 1).

Mathematica
```
Table[3^3^n+1,{n,0,6}]
```

a(n)=3^(3^n)+1 \\ Charles R Greathouse IV, Jul 05 2011

a(n) = 3^(3^n) + 1. a(n) = A055777(n) + 1.

A137840 Number of distinct n-ary operators in a quaternary logic.

Original entry on oeis.org

4, 256, 4294967296, 340282366920938463463374607431768211456, 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096

Offset: 0

Ross Drewe, Feb 13 2008

The total number of n-ary operators in a k-valued logic is T = k^(k^n), i.e. if S is a set of k elements, there are T ways of mapping an ordered subset of n elements taken from S to an element of S. Some operators are "degenerate": the operator has arity p, if only p of the n input values influence the output. Therefore the set of operators can be partitioned into n+1 disjoint subsets representing arities from 0 to n.

Cf. A001146 (in binary logic), A055777 (in a ternary logic), A137841 (in a quinternary logic).

Subsequence of A000302.

a(n) = 4^(4^n).

A140331 Smallest k such that 3^(3^n) - k is prime.

Original entry on oeis.org

1, 4, 2, 26, 10, 466, 506, 128, 13342, 5974, 8410, 6470

Offset: 0

Jonathan Vos Post, May 26 2008

This is to 3 as A058220 is to 2.

			a(4) = 10 because 3^(3^4) - 10 = 443426488243037769948249630619149892793 is prime.

Cf. A055777 (3^(3^n)), A058220.

a(n) = x = 3^(3^n); x - precprime(x); \\ Michel Marcus, Jan 28 2015

a(n) = A064722(A055777(n)). - Michel Marcus, Jan 28 2015

a(5)-a(8) from Cino Hilliard, Jun 02 2008
a(9)-a(10) from Donovan Johnson, Sep 13 2008
a(0) prepended and a(11) from Michael S. Branicky, Aug 23 2024

A336068 Numbers k such that the exponent of the highest power of 3 dividing k (A007949) is a divisor of k.

Original entry on oeis.org

3, 6, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 48, 51, 54, 57, 60, 66, 69, 72, 75, 78, 84, 87, 90, 93, 96, 102, 105, 108, 111, 114, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 156, 159, 165, 168, 174, 177, 180, 183, 186, 189, 192, 195, 198, 201, 204

Offset: 1

Amiram Eldar, Jul 07 2020

nonn

All the terms are divisible by 3 by definition.

Šalát (1994) proved that the asymptotic density of this sequence is 0.287106... (A336069).

			3 is a term since A007949(3) = 1 is a divisor of 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Tibor Šalát, On the function a_p, p^a_p(n) || n (n > 1), Mathematica Slovaca, Vol. 44, No. 2 (1994), pp. 143-151.

A055777 is a subsequence.
Cf. A007949, A336069.
Similar sequences: A033950, A005349, A006446, A074946, A075592, A007694, A112249, A336066.

Select[Range[200], Mod[#, 3] == 0 && Divisible[#, IntegerExponent[#, 3]] &]

isok(m) = if (!(m%3), (m % valuation(m,3)) == 0); \\ Michel Marcus, Jul 08 2020

A383817 Decimal expansion of -Sum_{k>=1} mu(3*k)/(3^k - 1), where mu is the Möbius function A008683.

Original entry on oeis.org

3, 7, 0, 4, 2, 1, 1, 7, 5, 6, 3, 3, 9, 2, 6, 7, 9, 8, 4, 9, 5, 7, 4, 3, 1, 8, 9, 4, 1, 1, 2, 6, 8, 1, 0, 0, 9, 7, 8, 1, 2, 8, 5, 9, 6, 7, 8, 4, 6, 0, 5, 3, 3, 4, 8, 1, 5, 3, 8, 8, 6, 0, 2, 7, 8, 1, 5, 4, 3, 8, 6, 7, 8, 3, 1, 5, 7, 3, 5, 1, 5, 6, 5, 6, 0, 1, 0

Offset: 0

Artur Jasinski, May 11 2025

The real root of the cubic polynomial 729*x^3 - 810*x^2 - 429*x + 233 matches this constant to 20 decimal places.

			0.3704211756339267984957431894112681...

Maxie Dion Schmidt, A catalog of interesting and useful Lambert series identities, arXiv:2004.02976 [math.NT], 2004(2020), page 9, eq. (3.1h), put q = 1/3 and alpha = 3. - _Amiram Eldar_, May 11 2025

Cf. A007404, A055777, A383818, A383819, A383820.

sum(k=1,logint(2^getlocalbitprec(),3)+1,moebius(3*k)/(3.^k - 1),0.) \\ Bill Allombert

Equals Sum_{k>=0} 1/3^(3^k) = Sum_{k>=0} 1/A055777(k). - Amiram Eldar, May 11 2025

A137841 Number of distinct n-ary operators in a quinternary logic.

Original entry on oeis.org

5, 3125, 298023223876953125, 2350988701644575015937473074444491355637331113544175043017503412556834518909454345703125

Offset: 0

Ross Drewe, Feb 13 2008

The total number of n-ary operators in a k-valued logic is T = k^(k^n), i.e. if S is a set of k elements, there are T ways of mapping an ordered subset of n elements taken from S to an element of S. Some operators are "degenerate": the operator has arity p, if only p of the n input values influence the output. = therefore the set of operators can be partitioned into n+1 disjoint subsets representing arities from 0 to n.

Cf. A001146 (in binary logic), A055777 (in ternary logic), A137840 (in quaternary logic).

Subsequence of A000351.

a(n) = 5^(5^n).

A129291 a(n) = 1 - 3^(3^n) + 9^(3^n).

Original entry on oeis.org

7, 703, 387400807, 58149737003032434092905183, 196627050475552913618075908526912116282660024455971729157367165907347241304007

Offset: 0

Alexander Adamchuk, Apr 08 2007

nonn

a(n) is the ratio of two consecutive base-3 Fermat numbers A129290(n) = 3^(3^n) + 1 = {4, 28, 19684, 7625597484988, ...}.

Cf. A129290 (3^(3^n) + 1).
Cf. A055777 (3^(3^n)).
Cf. A002061 (central polygonal numbers: n^2 - n + 1).

Mathematica
```
Table[1 - 3^3^n + 9^3^n, {n,0,5}]
```

a(n) = A002061(3^(3^n)). a(n) = A129290(n+1) / A129290(n).

A370578 Numbers k such that k + 1 divides 3^k + 1.

Original entry on oeis.org

0, 1, 3, 27, 531, 1035, 4635, 6363, 11475, 19683, 40131, 80955, 266475, 280755, 307395, 356643, 490371, 544347, 557955, 565515, 572715, 808227, 1256355, 1695483, 1959075, 1995075, 2771595, 2837835, 3004155, 3208491, 3337635, 3886443, 4670955, 5619411, 6434595, 6942817

Offset: 1

Akiva Weinberger, Feb 22 2024

nonn

The sequence is infinite. It contains all numbers of the form 3^(3^m).
After 3, the smallest term that is not a multiple of 9 is a(13) = 266475.
After 1, the smallest term that is not a multiple of 3 is a(36) = 6942817.
After 1, the smallest term that is not 3 (mod 8) is, also, a(36) = 6942817.
No term can be 2 (mod 3). Proof: Otherwise, k + 1 would be a multiple of 3 while 3^k + 1 would not.
All terms after 0 are odd. Proof: Suppose k is even, so that k+1 is odd. Let p be a prime factor of k+1. Then (by definition of k) 3^k == -1 (mod p) and 3^(2k) == 1 (mod p), so the order of 3 (mod p) divides 2k but not k. Thus the order of 3 is a multiple of 2^(v_2(k)+1) where v_2(k) = A007814(k) is the exponent of 2 in the prime factorization of k. But 3^(p-1) == 1 (mod p) by Fermat's little theorem, so p == 1 (mod 2^(v_2(k)+1)). Multiplying this for all prime factors p of k+1 gives k+1 == 1 (mod 2^(v_2(k)+1)), or k == 0 (mod 2^(v_2(k)+1)). But this contradicts the definition of v_2.
Apart from 0, the only possible residues mod 72 are 1, 3, 13, 25, 27, 37, 49, 51, and 61. It is conjectured that all appear eventually. (See John Omielan's answer to the author's question on Mathematics Stack Exchange.)
Empirically, approximately 80% of the terms are 27 (mod 72).

			(3^27+1)/(27+1) is an integer, so 27 is in the sequence. This can be shown efficiently using a modular exponentiation algorithm to find 3^27 mod 28.

Zubin Mukerjee, Table of n, a(n) for n = 1..214
Mathematics Stack Exchange, Why do so many solutions to n+1 | 3^n+1 satisfy n = 27 (mod 72)?

Cf. A006521, A006517, A007814, A015973, A055777.

Select[Range[0,7000000],TrueQ[PowerMod[3,#,#+1]==#]&] (* James C. McMahon, Feb 25 2024 *)

for n in range(100_000_000):
  if (pow(3,n,n+1)==n):
    print(n)

cp's OEIS Frontend

A364855 Initial digit of 3^(3^n) (A055777(n)).

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A129290 Base-3 Fermat numbers: 3^(3^n) + 1.

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A137840 Number of distinct n-ary operators in a quaternary logic.

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A140331 Smallest k such that 3^(3^n) - k is prime.

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A336068 Numbers k such that the exponent of the highest power of 3 dividing k (A007949) is a divisor of k.

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A383817 Decimal expansion of -Sum_{k>=1} mu(3*k)/(3^k - 1), where mu is the Möbius function A008683.

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A137841 Number of distinct n-ary operators in a quinternary logic.

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A129291 a(n) = 1 - 3^(3^n) + 9^(3^n).

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