cp's OEIS Frontend

Number of ways of writing n as a product of primes.

Number of ways of writing n as a sum of distinct powers of 2.

Continued fraction for golden ratio A001622.

Partial sums of A000007 (characteristic function of 0). - Jeremy Gardiner, Sep 08 2002

An example of an infinite sequence of positive integers whose distinct pairwise concatenations are all primes! - Don Reble, Apr 17 2005

Binomial transform of A000007; inverse binomial transform of A000079. - Philippe Deléham, Jul 07 2005

A063524(a(n)) = 1. - Reinhard Zumkeller, Oct 11 2008

For n >= 0, let M(n) be the matrix with first row = (n n+1) and 2nd row = (n+1 n+2). Then a(n) = absolute value of det(M(n)). - K.V.Iyer, Apr 11 2009

The partial sums give the natural numbers (A000027). - Daniel Forgues, May 08 2009

From Enrique Pérez Herrero, Sep 04 2009: (Start)

a(n) is also tau_1(n) where tau_2(n) is A000005.

a(n) is a completely multiplicative arithmetical function.

a(n) is both squarefree and a perfect square. See A005117 and A000290. (End)

Also smallest divisor of n. - Juri-Stepan Gerasimov, Sep 07 2009

Also decimal expansion of 1/9. - Enrique Pérez Herrero, Sep 18 2009; corrected by Klaus Brockhaus, Apr 02 2010

a(n) is also the number of complete graphs on n nodes. - Pablo Chavez (pchavez(AT)cmu.edu), Sep 15 2009

Totally multiplicative sequence with a(p) = 1 for prime p. Totally multiplicative sequence with a(p) = a(p-1) for prime p. - Jaroslav Krizek, Oct 18 2009

n-th prime minus phi(prime(n)); number of divisors of n-th prime minus number of perfect partitions of n-th prime; the number of perfect partitions of n-th prime number; the number of perfect partitions of n-th noncomposite number. - Juri-Stepan Gerasimov, Oct 26 2009

For all n>0, the sequence of limit values for a(n) = n!*Sum_{k>=n} k/(k+1)!. Also, a(n) = n^0. - Harlan J. Brothers, Nov 01 2009

a(n) is also the number of 0-regular graphs on n vertices. - Jason Kimberley, Nov 07 2009

Differences between consecutive n. - Juri-Stepan Gerasimov, Dec 05 2009

From Matthew Vandermast, Oct 31 2010: (Start)

1) When sequence is read as a regular triangular array, T(n,k) is the coefficient of the k-th power in the expansion of (x^(n+1)-1)/(x-1).

2) Sequence can also be read as a uninomial array with rows of length 1, analogous to arrays of binomial, trinomial, etc., coefficients. In a q-nomial array, T(n,k) is the coefficient of the k-th power in the expansion of ((x^q -1)/(x-1))^n, and row n has a sum of q^n and a length of (q-1)*n + 1. (End)

The number of maximal self-avoiding walks from the NW to SW corners of a 2 X n grid.

When considered as a rectangular array, A000012 is a member of the chain of accumulation arrays that includes the multiplication table A003991 of the positive integers. The chain is ... < A185906 < A000007 < A000012 < A003991 < A098358 < A185904 < A185905 < ... (See A144112 for the definition of accumulation array.) - Clark Kimberling, Feb 06 2011

a(n) = A007310(n+1) (Modd 3) := A193680(A007310(n+1)), n>=0. For general Modd n (not to be confused with mod n) see a comment on A203571. The nonnegative members of the three residue classes Modd 3, called [0], [1], and [2], are shown in the array A088520, if there the third row is taken as class [0] after inclusion of 0. - Wolfdieter Lang, Feb 09 2012

Let M = Pascal's triangle without 1's (A014410) and V = a variant of the Bernoulli numbers A027641 but starting [1/2, 1/6, 0, -1/30, ...]. Then M*V = [1, 1, 1, 1, ...]. - Gary W. Adamson, Mar 05 2012

As a lower triangular array, T is an example of the fundamental generalized factorial matrices of A133314. Multiplying each n-th diagonal by t^n gives M(t) = I/(I-t*S) = I + t*S + (t*S)^2 + ... where S is the shift operator A129184, and T = M(1). The inverse of M(t) is obtained by multiplying the first subdiagonal of T by -t and the other subdiagonals by zero, so A167374 is the inverse of T. Multiplying by t^n/n! gives exp(t*S) with inverse exp(-t*S). - Tom Copeland, Nov 10 2012

The original definition of the meter was one ten-millionth of the distance from the Earth's equator to the North Pole. According to that historical definition, the length of one degree of latitude, that is, 60 nautical miles, would be exactly 111111.111... meters. - Jean-François Alcover, Jun 02 2013

Deficiency of 2^n. - Omar E. Pol, Jan 30 2014

Consider n >= 1 nonintersecting spheres each with surface area S. Define point p on sphere S_i to be a "public point" if and only if there exists a point q on sphere S_j, j != i, such that line segment pq INTERSECT S_i = {p} and pq INTERSECT S_j = {q}; otherwise, p is a "private point". The total surface area composed of exactly all private points on all n spheres is a(n)*S = S. ("The Private Planets Problem" in Zeitz.) - Rick L. Shepherd, May 29 2014

For n>0, digital roots of centered 9-gonal numbers (A060544). - Colin Barker, Jan 30 2015

Product of nonzero digits in base-2 representation of n. - Franklin T. Adams-Watters, May 16 2016

Alternating row sums of triangle A104684. - Wolfdieter Lang, Sep 11 2016

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

Length of period of continued fraction for sqrt(A002522) or sqrt(A002496). - A.H.M. Smeets, Oct 10 2017

a(n) is also the determinant of the (n+1) X (n+1) matrix M defined by M(i,j) = binomial(i,j) for 0 <= i,j <= n, since M is a lower triangular matrix with main diagonal all 1's. - Jianing Song, Jul 17 2018

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = min(i,j) for 1 <= i,j <= n (see Xavier Merlin reference). - Bernard Schott, Dec 05 2018

a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = tau(gcd(i,j)) for 1 <= i,j <= n (see De Koninck & Mercier reference). - Bernard Schott, Dec 08 2020

Occasionally defined with a(0) = 0.

Conjecture: no matter what the starting term is, the sequence eventually joins this one. This should be true in any base - base 2, for example, is trivial.

A063114 iterated, beginning with 1. - Reinhard Zumkeller, Jan 15 2012

To determine whether a given number k is a term of this sequence, start with k, take the cube of the product of its nonzero digits, apply the same process to the result, and continue until 30 iterations are reached. If 1 is reached during the process, k is a term of this sequence. If not, k is not a term of this sequence.

Every power 10^k is a term of this sequence.

If k is a term, the numbers obtained by inserting zeros anywhere in k are terms.

If k is a term, the numbers obtained by inserting ones anywhere in k are terms.

If k is a term, each distinct permutation of the digits of k gives another term.

If k is a term, the number of iterations required to converge to 1 is less than or equal to 10 (conjectured).

From Antti Karttunen, Nov 17 2016: (Start)

This is a filter-sequence for decimal base: a(n) = the least number with the same prime signature as A054842(n).

This sequence can be used for filtering certain base-10 related sequences, because it matches only with any such sequence b that can be computed as b(n) = f(A054842(n)), where f(n) is any function that depends only on the prime signature of n (some of these are listed under the index entry for "sequences computed from exponents in ...").

Matching in this context means that the sequence a matches with the sequence b iff for all i, j: a(i) = a(j) => b(i) = b(j). In other words, iff the sequence b partitions the natural numbers to the same or coarser equivalence classes (as/than the sequence a) by the distinct values it obtains.

Any such sequence should match where the result is computed from the nonzero decimal digits of n, but does not depend on their order. These include for example, A007953 (digital sum and any of its variants), A010888 (digital root of n) and A051801 (product of the nonzero digits of n). As of Nov 11 2016, there were a couple of hundred such sequences that seemed to match with this one. These are given at the "List of sequences whose equivalence classes ..." link.

(End)

The graph somewhat resembles wisteria flowers.