cp's OEIS Frontend

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

Except for a(2)=2, all the terms in the sequence are odd. This is because of the existence of a non-cyclic dihedral group of order 2n for each n>1. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 09 2001

Also gcd(n, A051953(n)) = 1. - Labos Elemer

n such that x^n == 1 (mod n) has no solution 2 <= x <= n. - Benoit Cloitre, May 10 2002

There is only one group (the cyclic group of order n) whose order is n. - Gerard P. Michon, Jan 08 2008 [This is a 1947 result of Tibor Szele. - Charles R Greathouse IV, Nov 23 2011]

Any divisor of a Carmichael number (A002997) must be odd and cyclic. Conversely, G. P. Michon conjectured (c. 1980) that any odd cyclic number has at least one Carmichael multiple (if the conjecture is true, each of them has infinitely many such multiples). In 2007, Michon & Crump produced explicit Carmichael multiples of all odd cyclic numbers below 10000 (see link, cf. A253595). - Gerard P. Michon, Jan 08 2008

Numbers n such that phi(n)^phi(n) == 1 (mod n). - Michel Lagneau, Nov 18 2012

Contains A000040, and all members of A006094 except 6. - Robert Israel, Jul 08 2015

Number m such that n^n == r (mod m) is solvable for any r. - David W. Wilson, Oct 01 2015

Numbers m such that A074792(m) = m + 1. - Thomas Ordowski, Jul 16 2017

Squarefree terms of A056867 (see McCarthy link p. 592 and similar comment with "cubefree" in A051532). - Bernard Schott, Mar 24 2022

It is interesting that most of the numbers have the last digit 1. For example 530881, 3581761, 7207201, etc.

Granville & Pomerance conjecture that there are ~ c x^(1/3)/(log x)^3 terms of this sequence up to x. Heath-Brown proves that, for any e > 0, there are O(x^(7/20 + e)) terms of this sequence up to x. - Charles R Greathouse IV, Nov 19 2012

All 3-term Carmichael numbers can be represented by certain Chernick polynomials, whose values obey a strict s-decomposition (A324460) besides certain exceptions. Under Dickson's conjecture, "almost all" 3-term Carmichael numbers are primary Carmichael numbers (A324316) in the sense that C'3(x)/C_3(x) -> 1 as x -> infinity, where C_3(x) counts the 3-term Carmichael numbers and C'_3(x) counts the 3-term primary Carmichael numbers up to x. A primary Carmichael number m has the unique property (*) that for each prime divisor p of m, the sum of the base-p digits of m equals p. As a nontrivial result, all 3-term Carmichael numbers m have at least the property that (*) holds for the greatest prime divisor p of m, see Kellner 2019. - _Bernd C. Kellner, Aug 03 2022

GCD(b,n)=1 and b^(n+1) == 1 (mod n).

The sequence lists the squarefree composite numbers n such that every prime divisor p of n satisfies (p-1)|(n+1) (similar to Korselt's criterion).

The sequence can be considered as an extension of k-Knödel numbers to k negative, in this case equal to -1.

Numbers n > 3 such that b^(n+2) == b (mod n) for every integer b. Also, numbers n > 3 such that A002322(n) divides n+1. Are there infinitely many such numbers? It seems that such numbers n > 35 have at least three prime factors. - Thomas Ordowski, Jun 25 2017

Proof that 15 and 35 are the only numbers with only two prime factors (and so all others have >= 3): Since n is squarefree and composite, it has at least two prime factors, p and q. If these are the only two, n = p*q. Then the criterion is that (p-1)|(n+1) -> (p-1)|pq+1, and q-1|pq+1. Write pq+1 = j*(p-1) = k*(q-1). Rearranging, p*(j-q)=j+1 and q*(k-p)=k+1. Since j = (pq+1)/(p-1), for large n, j~=q, and k~=p. But we see that p divides j+1~=q, and q divides k+1~=p. For large n this is only possible if p and q are roughly equal, so j-q=k-p=1. Otherwise, we have j+1 >= 2*p and k+1 >= 2*q, and which puts upper bounds on p and q. Enumerating these gives (p,q)=(3,5) and (p,q)=(5,7) as the only solutions. - Alex Meiburg, Oct 03 2024

The GCD of integers x^(n+1)-x, for all integers x. - Roger Cuculiere (cuculier(AT)imaginet.fr), Jan 19 2002

If each x in a ring satisfies x^(n+1)=x, the characteristic of the ring is a divisor of a(n) (Rosenblum 1977). - Daniel M. Rosenblum (DMRosenblum(AT)world.oberlin.edu), Sep 24 2008

The denominators of the Bernoulli numbers for n>0. B_n sequence begins 1, -1/2, 1/6, 0/2, -1/30, 0/2, 1/42, 0/2, ... This is an alternative version of A027642 suggested by the theorem of Clausen. To add a(0) = 1 has been proposed in A141056. - Peter Luschny, Apr 29 2009

For N > 1, a(n) is the greatest number k such that x*y^n ==y*x^n (mod k) for any integers x and y. Example: a(19) = 798 because x*y^19 ==y*x^19 (mod 798). - Michel Lagneau, Apr 21 2012

a(n) is the largest k such that b^(n+1) == b (mod k) for every integer b. - Mateusz Szymański, Feb 18 2016, corrected by Thomas Ordowski, Jul 01 2018

When n is even, a(n) is the product of the distinct primes dividing the denominator of zeta(1-n), where zeta(s) is the Riemann zeta function. - Griffin N. Macris, Jun 13 2016

If n+1 is prime, then A002322(a(n)) = n. Composite numbers n+1 such that A002322(a(n)) = n are in A317210. - Max Alekseyev and Thomas Ordowski, Jul 09 2018

Includes 341 which is first pseudoprime to base 2 and distinguishes sequence from A008578.

First composite even term is a(14868) = 161038 = A006935(2). - Max Alekseyev, Feb 11 2015

If k is a term, then so is 2^k - 1. - Max Alekseyev, Sep 22 2016

Terms of the form 2^k - 2 correspond to k in A296104. - Max Alekseyev, Dec 04 2017

If 2^k - 1 is a term, then so is k. - Thomas Ordowski, Apr 27 2018

Alford, Grantham, Hayman, & Shallue construct large Carmichael numbers, finding upper bounds for a(3)-a(19565220) and a(10333229505). - Charles R Greathouse IV, May 30 2013

a(n) = number of bases b modulo n for which b^{n-1} == 1 (mod n).

a(A209211(n)) = 1. - Reinhard Zumkeller, Mar 02 2013

A049559(n) divides a(n) divides A000010(n). - Thomas Ordowski, Dec 14 2013

Note that a(n) = phi(n) iff n = 1 or n is prime or n is Carmichael number A002997. - Thomas Ordowski, Dec 17 2013

The product is finite, as the sum of the base-p digits of n is n if p > n.

a(198) = 2465 is the only term below 10^6 that is a Carmichael number (A002997).

It appears that a(n)=1 if and only if n is in A094960. - Robert Israel, Mar 30 2020

It turns out that a(n) equals the denominator of the first derivative of the Bernoulli polynomial B(n,x). So a(n)=1 if and only if n is in A094960, also impyling that n+1 is prime. A324370 is also involved in such formulas regarding higher derivatives. See Kellner 2023. - Bernd C. Kellner, Oct 12 2023

All terms belong to A038509 (Composite numbers with smallest prime factor >= 5). Many but not all terms belong to A060976 (Odd nonprimes, c, which divide Bernoulli(2*c)).

Many terms are semiprimes:

- the non-semiprimes are {275, 455, 475, 539, 575, 715, 775, 875, 935, ...}: see A321487;

- semiprime terms that are multiples of 5 have indices {7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, ...} = A002145 (Primes of form 4*k + 3, except 3, or k > 0; or Primes which are also Gaussian primes);

- semiprime terms that are multiples of 7 have indices {5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, ...} = A003627 (Primes of form 3*k - 1, except 2, or k > 1);

- semiprime terms that are multiples of 11 have indices {5, 7, 13, 17, 19, 23, 37, 43, 47, 53, 59, 67, 73, 79, 83, ...} = Primes of the form 4*k + 1 and 4*k - 1. [Edited by Michel Marcus, Jul 21 2018, M. F. Hasler, Nov 09 2018]

Conjecture: odd numbers n > 1 such that n divides Sum_{k=1..n-1} k^(n-1). - Thomas Ordowski and Robert Israel, Oct 09 2015. Professor Andrzej Schinzel (in a letter to me, dated Dec 29 2015) proved this conjecture. - Thomas Ordowski, Jul 20 2018

Note that n^2 divides Sum_{k=1..n-1} k^n for every odd number n > 1. - Thomas Ordowski, Oct 30 2015

Conjecture: these are "anti-Carmichael numbers" defined; n > 1 such that p - 1 does not divide n - 1 for every prime p dividing n. Equivalently, odd numbers n > 1 such that n is coprime to A027642(n-1). A number n > 1 is an "anti-Carmichael" if and only if gcd(n, b^n - b) = 1 for some integer b. - Thomas Ordowski, Jul 20 2018

It seems that these numbers are all composite terms of A317358. - Thomas Ordowski, Jul 30 2018

a(62) = 697 is the first term not in A267999: see A306097 for all these terms. - M. F. Hasler, Nov 09 2018

If the conjecture from Thomas Ordowski is true, then no term is a multiple of 2 or 3. - Jianing Song, Jan 27 2019

Conjecture: an odd number n > 1 is a term iff gcd(n, A027642(n-1)) = 1. - Thomas Ordowski, Jul 19 2019

Conjecture: Sequence consists of numbers n > 1 such that r = b^n + n - b will produce a prime for one or more integers b > 1. Only when n is in this sequence do one or more prime factors of n fail to divide r for all b. Also, n and b must be coprime for r to be prime. The above also applies to r = b^n - n - b, ignoring n=3, b=2. - Richard R. Forberg, Jul 18 2020

Odd numbers n > 1 such that Sum_{k(even)=2..n-1}2*k^(n-1) == 0 (mod n). - Davide Rotondo, Oct 28 2020

What is the asymptotic density of these numbers? The numbers A267999 have a slightly lower density. The difference between the densities is equal to the density of the numbers A306097. - Thomas Ordowski, Feb 15 2021

The asymptotic density of this sequence is in the interval (0.253, 0.265) (Ordowski, 2021). - Amiram Eldar, Feb 26 2021