cp's OEIS Frontend

The prime p divides 3^p - 2^p - 1. Quotients (3^p - 2^p - 1)/p, where p = Prime[n], are listed in A127071. - Alexander Adamchuk, Jan 31 2008

a(7) > 10^9. [From D. S. McNeil, Mar 16 2009]

a(n+1) is the sum of the elements in the n-th row of triangle pertaining to A036561. - Amarnath Murthy, Jan 02 2002

Number of 2 X n binary arrays with a path of adjacent 1's and no path of adjacent 0's from top row to bottom row. - R. H. Hardin, Mar 21 2002

With offset 1, partial sums of A027649. - Paul Barry, Jun 24 2003

Number of distinct lines through the origin in the n-dimensional lattice of side length 2. A049691 has the values for the 2-dimensional lattice of side length n. - Joshua Zucker, Nov 19 2003

a(n+1)/(n+1)=(3*3^n-2*2^n)/(n+1) is the second binomial transform of the harmonic sequence 1/(n+1). - Paul Barry, Apr 19 2005

a(n+1) is the sum of n-th row of A036561. - Reinhard Zumkeller, May 14 2006

The sequence gives the sum of the lengths of the segments in Cantor's dust generating sequence up to the i-th step. Measurement unit = length of the segment of i-th step. - Giorgio Balzarotti, Nov 18 2006

Let T be a binary relation on the power set P(A) of a set A having n = |A| elements such that for every element x, y of P(A), xTy if x is a proper subset of y. Then a(n) = |T|. - Ross La Haye, Dec 22 2006

From Alexander Adamchuk, Jan 04 2007: (Start)

a(n) is prime for n in A057468.

p divides a(p) - 1 for prime p.

Quotients (3^p - 2^p - 1)/p, where p = prime(n), are listed in A127071.

Numbers k such that k divides 3^k - 2^k - 1 are listed in A127072.

Pseudoprimes in A127072(n) include all powers of primes {2,3,7} and some composite numbers that are listed in A127073, which includes all Carmichael numbers A002997.

Numbers n such that n^2 divides 3^n - 2^n - 1 are listed in A127074.

5 divides a(2n).

5^2 divides a(2*5n).

5^3 divides a(2*5^2n).

5^4 divides a(2*5^3n).

7^2 divides a(6*7n).

13 divides a(4n).

13^2 divides a(4*13n).

19 divides a(3n).

19^2 divides a(3*19n).

23^2 divides a(11n).

23^3 divides a(11*23n).

23^4 divides a(11*23^2n).

29 divides a(7n).

p divides a((p-1)n) for prime p>3.

p divides a((p-1)/2) for prime p in A097934. Also primes p such that 6 is a square mod p, except {2,3}, A038876(n).

p^(k+1) divides a(p^k*(p-1)/2*n) for prime p in A097934.

p^(k+1) divides a(p^k*(p-1)*n) for prime p>3.

Note the exception that for p = 23, p^(k+2) divides a(p^k*(p-1)/2*n).

There are no more such exceptions for primes p up to 600000. (End)

a(n) divides a(q*(n+1)-1), for all q integer. Leonardo Sarasua, Apr 15 2024

Final digits of terms follow sequence 1,5,9,5. - Enoch Haga, Nov 26 2007

This is also the second column sequence of the Sheffer triangle A143494 (2-restricted Stirling2 numbers). See the e.g.f. given below. - Wolfdieter Lang, Oct 08 2011

Partial sums give A000392. - Jon Perry, Apr 05 2014

For n >= 1, this is also row 2 of A281890: when consecutive positive integers are written as a product of primes in nondecreasing order, "3" occurs in n-th position a(n) times out of every 6^n. - Peter Munn, May 17 2017

a(n) is the number of ternary sequences of length n which include the digit 2. For example, a(2)=5 since the sequences are 02,20,12,21,22. - Enrique Navarrete, Apr 05 2021

a(n-1) is the number of ways we can form disjoint unions of two nonempty subsets of [n] such that the union contains n. For example, for n = 3, a(2) = 5 since the disjoint unions are {1}U{3}, {1}U{2,3}, {2}U{3}, {2}U{1,3}, and {1,2}U{3}. Cf. A000392 if we drop the requirement that the union contains n. - Enrique Navarrete, Aug 24 2021

Configures as a composite Koch Snowflake Fractal (see illustration in links) based on the five-fold division of the Cantor Square/Cantor Dust Fractal of (9^n-4^n)/5 see my illustration in (A016153). - John Elias, Oct 13 2021

Number of pairs (A,B) where B is a subset of {1,2,...,n} and A is a proper subset of B. - Jianing Song, Jun 18 2022

From Manfred Boergens, Mar 29 2023: (Start)

With regard to the comments by Ross La Haye and Jianing Song: Omitting "proper" gives A000244.

Number of pairs (A,B) where B is a nonempty subset of {1,2,...,n} and A is a nonempty subset of B. For nonempty proper subsets see a(n+1) in A028243. (End)

a(n) is the number of n-digit numbers whose smallest decimal digit is 7. - Stefano Spezia, Nov 15 2023

a(n-1) is the number of all possible player-reduced binary games observed by each player in an nx2 game assuming the individual strategies of k < n - 1 players are fixed and the remaining n - k - 1 player will play as one, either maintaining their status quo strategies or jointly adopting an alternative strategy. - Ambrosio Valencia-Romero, Apr 11 2024

Prime p divides 3^p - 2^p - 1. 42 = 2*3*7 divides a(n) for n>2.

Numbers n such that n divides 3^n - 2^n - 1 are listed in A127072.

Pseudoprimes in A127072 include all powers of primes {2,3,7} and some composite numbers that are listed in A127073.

Numbers n such that n^2 divides 3^n - 2^n - 1 are listed in A127074.

Numbers n such that n^3 divides 3^n - 2^n - 1 are {1,4,7,...}.

Prime p divides 3^p - 2^p - 1.

Quotients (3^p - 2^p - 1)/p, where p is prime, are listed in A127071.

Pseudoprimes in a(n) include all powers of primes {2,3,7} and some composite numbers that are listed in A127073.

Numbers k such that k^2 divides 3^k - 2^k - 1 are listed in A127074.

Numbers k such that k^3 divides 3^k - 2^k - 1 are {1, 4, 7, ...}.

This sequence includes all the Carmichael numbers (A002997).

Prime p divides 3^p - 2^p - 1. Quotients (3^p - 2^p - 1)/p, where p = prime(n), are listed in A127071.

Numbers k such that k divides 3^k - 2^k - 1 are listed in A127072.

Pseudoprimes in A127072 include all the powers of the primes {2, 3, 7}.

Numbers k such that k^2 divides 3^k - 2^k - 1 are listed in A127074.

Numbers k such that k^3 divides 3^k - 2^k - 1 are 1, 4, 7, ...

The perfect squares in listed terms are a(1) = 1, a(3) = 49 = 7^2, a(13) = 32041 = 179^2 and a(29) = 383161 = 619^2.

Note that primes {7,179,619} are the terms of A130060 or primes in A127074.

It appears that all terms are composite except a(1) = 1 and a(2) = 3. Most listed terms are divisible by 3, except {1, 35, 583, 70643, ...}.

Primes = 1 or 23 mod 24. Hence, together with 2, primes such that (2/p) = 1 = (3/p) where (k/p) is the Legendre symbol. - Charles R Greathouse IV, Apr 06 2012

A000040(n) such that A260531(n) = 1.

Is this a subsequence of A130060?

a(6) > 10325801 if it exists.

a(6) > 3037000499 if it exists. - Hiroaki Yamanouchi, Aug 20 2015

If it exists, a(41) > 10^9.

Of the 4*A015616(10) = 4*109 = 436 integer sequences of one of the forms

Numbers k such that k^2 | A^k + B^k + C^k,

Numbers k such that k^2 | A^k + B^k - C^k,

Numbers k such that k^2 | A^k - B^k + C^k,

or Numbers k such that k^2 | A^k - B^k - C^k

such that 0 < C < B < A <= 10 and gcd(A,B,C)=1, this one appears to have the largest number of terms.

By comparison, A127074 (k such that k^2 | 3^k - 2^k - 1) and A343115 (k such that k^2 | 5^k - 3^k - 2^k) seem unlikely to have any terms beyond A127074(9)=17807 and A343115(14)=876, respectively. Only 25 of the 436 above sequences have any 4-, 5-, or 6-digit terms at all.

a(41) > 10^11 if it exists. - Chai Wah Wu, May 16 2021