cp's OEIS Frontend

Partial sum of A055270.

Conjecture in "Introduction à la théorie des nombres" by J. M. Deconinck and Armel Mercier: this is the period length of the fraction 1/7^n. For example 1/7^2=0.0204081632653061224489795918367346938775510204....with a period of 42 digits =6*7=a(2). The period of 1/7^3 has exactly 294=a(3) digits. - Benoit Cloitre, Feb 02 2002

Also phi(7^n), where phi is Euler's totient function. - Alonso del Arte, May 08 2006

For n>=1, a(n) is equal to the number of functions f:{1,2...,n}->{1,2,3,4,5,6,7} such that for a fixed x in {1,2,...,n} and a fixed y in {1,2,3,4,5,6,7} we have f(x)<>y. - Aleksandar M. Janjic and Milan Janjic, Mar 27 2007

a(n) is the number of compositions of n when there are 6 types of each part. - Milan Janjic, Aug 13 2010

Apart from the first term, number of monic squarefree polynomials over F_7 of degree n. - Charles R Greathouse IV, Feb 07 2012

This sequence appears in the solution of the curvature sequence of the touching circle and chord example given by Kival Ngaokrajang in A249458. See also the pair A249864(n) and a(n-1), with a(-1) = 0, for which details are given in A249864.

Changing the offset to 1 gives the arithmetical function a(1) = 1, a(n) = 0 for n > 1, the identity function for Dirichlet multiplication (see Apostol). - N. J. A. Sloane

Changing the offset to 1 makes this the decimal expansion of 1. - N. J. A. Sloane, Nov 13 2014

Hankel transform (see A001906 for definition) of A000007 (powers of 0), A000012 (powers of 1), A000079 (powers of 2), A000244 (powers of 3), A000302 (powers of 4), A000351 (powers of 5), A000400 (powers of 6), A000420 (powers of 7), A001018 (powers of 8), A001019 (powers of 9), A011557 (powers of 10), A001020 (powers of 11), etc. - Philippe Deléham, Jul 07 2005

This is the identity sequence with respect to convolution. - David W. Wilson, Oct 30 2006

a(A000004(n)) = 1; a(A000027(n)) = 0. - Reinhard Zumkeller, Oct 12 2008

The alternating sum of the n-th row of Pascal's triangle gives the characteristic function of 0, a(n) = 0^n. - Daniel Forgues, May 25 2010

The number of maximal self-avoiding walks from the NW to SW corners of a 1 X n grid. - Sean A. Irvine, Nov 19 2010

Historically there has been some disagreement as to whether 0^0 = 1. Graphing x^0 seems to support that conclusion, but graphing 0^x instead suggests that 0^0 = 0. Euler and Knuth have argued in favor of 0^0 = 1. For some calculators, 0^0 triggers an error, while in Mathematica, 0^0 is Indeterminate. - Alonso del Arte, Nov 15 2011

Another consequence of changing the offset to 1 is that then this sequence can be described as the sum of Moebius mu(d) for the divisors d of n. - Alonso del Arte, Nov 28 2011

With the convention 0^0 = 1, 0^n = 0 for n > 0, the sequence a(n) = 0^|n-k|, which equals 1 when n = k and is 0 for n >= 0, has g.f. x^k. A000007 is the case k = 0. - George F. Johnson, Mar 08 2013

A fixed point of the run length transform. - Chai Wah Wu, Oct 21 2016

Sometimes called sopf(n).

Sum of primes dividing n (without repetition) (compare A001414).

Equals A051731 * A061397 = inverse Mobius transform of [0, 2, 3, 0, 5, 0, 7, ...]. - Gary W. Adamson, Feb 14 2008

Equals row sums of triangle A143535. - Gary W. Adamson, Aug 23 2008

a(n) = n if and only if n is prime. - Daniel Forgues, Mar 24 2009

a(n) = n is a new record if and only if n is prime. - Zak Seidov, Jun 27 2009

a(A001043(n)) = A191583(n);

For n > 0: a(A000079(n)) = 2, a(A000244(n)) = 3, a(A000351(n)) = 5, a(A000420(n)) = 7;

a(A006899(n)) <= 3; a(A003586(n)) = 5; a(A033846(n)) = 7; a(A033849(n)) = 8; a(A033847(n)) = 9; a(A033850(n)) = 10; a(A143207(n)) = 10. - Reinhard Zumkeller, Jun 28 2011

For n > 1: a(n) = Sum(A027748(n,k): 1 <= k <= A001221(n)). - Reinhard Zumkeller, Aug 27 2011

If n is the product of twin primes (A037074), a(n) = 2*sqrt(n+1) = sqrt(4n+4). - Wesley Ivan Hurt, Sep 07 2013

From Wilf A. Wilson, Jul 21 2017: (Start)

a(n) + 2, n > 2, is the number of maximal subsemigroups of the monoid of orientation-preserving or -reversing mappings on a set with n elements.

a(n) + 3, n > 2, is the number of maximal subsemigroups of the monoid of orientation-preserving or -reversing partial mappings on a set with n elements.

(End)

The smallest m such that a(m) = n, or 0 if no such number m exists is A064502(n). The only integers that are not in the sequence are 1, 4 and 6. - Bernard Schott, Feb 07 2022

Same as Pisot sequences E(1, 8), L(1, 8), P(1, 8), T(1, 8). Essentially same as Pisot sequences E(8, 64), L(8, 64), P(8, 64), T(8, 64). See A008776 for definitions of Pisot sequences.

If X_1, X_2, ..., X_n is a partition of the set {1..2n} into blocks of size 2 then, for n>=1, a(n) is equal to the number of functions f : {1..2n} -> {1,2,3} such that for fixed y_1,y_2,...,y_n in {1,2,3} we have f(X_i)<>{y_i}, (i=1..n). - Milan Janjic, May 24 2007

This is the auto-convolution (convolution square) of A059304. - R. J. Mathar, May 25 2009

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=1, a(n) equals the number of 8-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

a(n) is equal to the determinant of a 3 X 3 matrix with rows 2^(n+2), 2^(n+1), 2^n; 2^(n+3), 2^(n+4), 2(n+3); 2^n, 2^(n+1), 2^(n+2) when it is divided by 144. - J. M. Bergot, May 07 2014

a(n) gives the number of small squares in the n-th iteration of the Sierpinski carpet fractal. Equivalently, the number of vertices in the n-Sierpinski carpet graph. - Allan Bickle, Nov 27 2022

The binary representation of a(n) shows which prime numbers divide n, but not the multiplicities. a(2)=1, a(3)=10, a(4)=1, a(5)=100, a(6)=11, a(10)=101, a(30)=111, etc.

For n > 1, a(n) gives the (one-based) index of the column where n is located in array A285321. A008479 gives the other index. - Antti Karttunen, Apr 17 2017

From Antti Karttunen, Jun 18 & 20 2017: (Start)

A268335 gives all n such that a(n) = A248663(n); the squarefree numbers (A005117) are all the n such that a(n) = A285330(n) = A048675(n).

For all n > 1 for which the value of A285331(n) is well-defined, we have A285331(a(n)) <= floor(A285331(n)/2), because then n is included in the binary tree A285332 and a(n) is one of its ancestors (in that tree), and thus must be at least one step nearer to its root than n itself.

Conjecture: Starting at any n and iterating the map n -> a(n), we will always reach 0 (see A288569). This conjecture is equivalent to the conjecture that at any n that is neither a prime nor a power of two, we will eventually hit a prime number (which then becomes a power of two in the next iteration). If this conjecture is false then sequence A285332 cannot be a permutation of natural numbers. On the other hand, if the conjecture is true, then A285332 must be a permutation of natural numbers, because all primes and powers of 2 occur in definite positions in that tree. This conjecture also implies the conjectures made in A019565 and A285320 that essentially claim that there are neither finite nor infinite cycles in A019565.

If there are any 2-cycles in this sequence, then both terms of the cycle should be present in A286611 and the larger one should be present in A286612.

(End)

Binary rank of the distinct prime indices of n, where the binary rank of an integer partition y is given by Sum_i 2^(y_i-1). For all prime indices (with multiplicity) we have A048675. - Gus Wiseman, May 25 2024

No additional terms up to 20000. - Harvey P. Dale, Oct 02 2013

a(n)=2 iff n=2^k (k>0); otherwise a(n) is odd. If p is prime, a(p)=p; the converse is false, e.g.: a(15)=15. It is remarkable that this sequence often coincides with A006530, the largest prime P dividing n. Theorem: a(n)=P if and only if for every prime p < P in n there is some prime q in n with p|(q-1). - Franz Vrabec, Aug 30 2005

Probably finite. Is 378818692265664781682717625943 the largest term?

No further terms up to 7^50,000, a number with 42,255 digits. - Harvey P. Dale, Jul 14 2022

If the array is transposed, T(n,k) is the number of oriented rows of n colors using up to k different colors. The formula would be T(n,k) = [n==0] + [n>0]*k^n. The generating function for column k would be 1/(1-k*x). For T(3,2)=8, the rows are AAA, AAB, ABA, ABB, BAA, BAB, BBA, and BBB. - Robert A. Russell, Nov 08 2018

T(n,k) is the number of multichains of length n from {} to [k] in the Boolean lattice B_k. - Geoffrey Critzer, Apr 03 2020