cp's OEIS Frontend

a(n) depends only on the prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3 * 3 and 375 = 3 * 5^3 both have prime signature (3,1).

Multinomial coefficients in prime factorization order. - Max Alekseyev, Nov 07 2006

The Dirichlet inverse is given by A080339, negating all but the A080339(1) element in A080339. - R. J. Mathar, Jul 15 2010

Number of (distinct) permutations of the multiset of prime factors. - Joerg Arndt, Feb 17 2015

Number of not divisible chains in the divisor lattice of n. - Peter Luschny, Jun 15 2013

Note the distinctions between this and "n has exactly three prime factors" (A014612) or "n has exactly three distinct prime factors." (A033992). The word "sphenic" also means "shaped like a wedge" [American Heritage Dictionary] as in dentation with "sphenic molars." - Jonathan Vos Post, Sep 11 2005

Also the volume of a sphenic brick. A sphenic brick is a rectangular parallelepiped whose sides are components of a sphenic number, namely whose sides are three distinct primes. Example: The distinct prime triple (3,5,7) produces a 3x5x7 unit brick which has volume 105 cubic units. 3-D analog of 2-D A037074 Product of twin primes, per Cino Hilliard's comment. Compare with 3-D A107768 Golden 3-almost primes = Volumes of bricks (rectangular parallelepipeds) each of whose faces has golden semiprime area. - Jonathan Vos Post, Jan 08 2007

Sum(n>=1, 1/a(n)^s) = (1/6)*(P(s)^3 - P(3*s) - 3*(P(s)*P(2*s)-P(3*s))), where P is prime zeta function. - Enrique Pérez Herrero, Jun 28 2012

Also numbers n with A001222(n)=3 and A001221(n)=3. - Enrique Pérez Herrero, Jun 28 2012

n = 265550 is the smallest n with a(n) (=1279789) < A006881(n) (=1279793). - Peter Dolland, Apr 11 2020

Also, number of sequences of positive integers b_1, b_2, ..., b_k such that 1 + b_1*(1 + b_2*(...(1 + b_k) ... )) = n. If you take mu(b_1)*mu(b_2)*...*mu(b_k) for each sequence you get 1's 0's and -1's. Add them up and you get the terms for A007554. - Christian G. Bower, Oct 15 1998

Note that this applies also to planar rooted trees and other similar objects (mountain ranges, parenthesizations) encoded by A014486. - Antti Karttunen, Sep 07 2000

Equals sum of (n-1)-th row terms of triangle A152434. - Gary W. Adamson, Dec 04 2008

Equals the eigensequence of A051731, the inverse binomial transform. - Gary W. Adamson, Dec 26 2008

From Emeric Deutsch, Aug 18 2012: (Start)

The considered rooted trees are called generalized Bethe trees; in the Goldberg-Livshitz reference they are called uniform trees.

Also, a(n) = number of partitions of n-1 in which each part is divisible by the next. Example: a(5)=5 because we have 4, 31, 22, 211, and 1111.

There is a simple bijection between generalized Bethe trees with n+1 vertices and partitions of n in which each part is divisible by the next (the parts are given by the number of edges at the successive levels). We have the correspondences: number of edges --- sum of parts; root degree --- last part; number of leaves --- first part; height --- number of parts. (End)

a(n+1) = a(n) + 1 if and only if n is prime. - Jon Perry, Nov 24 2012

According to the MathOverflow link, log(a(n)) ~ log(4)*log(n)^2, and a more precise asymptotic expansion is similar to that of A018819 and hence A000123, so the conjecture in the Formula section is partly correct. - Andrey Zabolotskiy, Jan 22 2017

Also, a(n) = number of "non-squashing" partitions of 2n (or 2n+1), that is, partitions 2n = p_1 + p_2 + ... + p_k with 1 <= p_1 <= p_2 <= ... <= p_k and p_1 + p_2 + ... + p_i <= p_{i+1} for all 1 <= i < k [Hirschhorn and Sellers].

Row sums of A101566. - Paul Barry, Jan 03 2005

Equals infinite convolution product of [1,2,2,2,2,2,2,2,2] aerated A000079 - 1 times, i.e., [1,2,2,2,2,2,2,2,2] * [1,0,2,0,2,0,2,0,2] * [1,0,0,0,2,0,0,0,2]. - Mats Granvik and Gary W. Adamson, Aug 04 2009

Which can be further decomposed to the infinite convolution product of finally supported sequences, namely [1,1] aerated A000079 - 1 times with multiplicity A000027 + 1 times, i.e., [1,1] * [1,1] * [1,0,1] * [1,0,1] * [1,0,1] * ... (next terms are [1,0,0,0,1] 4 times, etc.). - Eitan Y. Levine, Jun 18 2023

Given A018819 = A000123 with repeats, polcoeff (1, 1, 2, 2, 4, 4, ...) * (1, 1, 1, ...) = (1, 2, 4, 6, 10, ...) = (1, 0, 2, 0, 4, 0, 6, ...) * (1, 2, 2, 2, ...). - Gary W. Adamson, Dec 16 2009

Let M = an infinite lower triangular matrix with (1, 2, 2, 2, ...) in every column shifted down twice. A000123 = lim_{n->infinity} M^n, the left-shifted vector considered as a sequence. Replacing (1, 2, 2, 2, ...) with (1, 3, 3, 3, ...) and following the same procedure, we obtain A171370: (1, 3, 6, 12, 18, 30, 42, 66, 84, 120, ...). - Gary W. Adamson, Dec 06 2009

First differences of the sequence are (1, 2, 2, 4, 4, 6, 6, 10, ...), A018819, i.e., the sequence itself with each term duplicated except for the first one (unless a 0 is prefixed before taking the first differences), as shown by the formula a(n) - a(n-1) = a(floor(n/2)), valid for all n including n = 0 if we let a(-1) = 0. - M. F. Hasler, Feb 19 2019

Sum over k <= n of number of partitions of k into powers of 2, A018819. - Peter Munn, Feb 21 2020

By a result of Karhumaki and Lifshits, this is also the number of polynomials p(x) with coefficients in {0,1} that divide x^n-1 and such that (x^n-1)/ {(x-1)p(x)} has all coefficients in {0,1}.

The number of tiles of a discrete interval of length n (an interval of Z). - Eric H. Rivals (rivals(AT)lirmm.fr), Mar 13 2007

Bodini and Rivals proved this is the number of tiles of a discrete interval of length n and also is the number (A107067) of polynomials p(x) with coefficients in {0,1} that divide x^n-1 and such that (x^n-1)/ {(x-1)p(x)} has all coefficients in {0,1} (Bodini, Rivals, 2006). This structure of such tiles is also known as Krasner's factorization (Krasner and Ranulac, 1937). The proof also gives an algorithm to recognize if a set is a tile in optimal time and in this case, to compute the smallest interval it can tile (Bodini, Rivals, 2006). - Eric H. Rivals (rivals(AT)lirmm.fr), Mar 13 2007

Number of lone-child-avoiding rooted achiral (or generalized Bethe) trees with positive integer leaves summing to n, where a rooted tree is lone-child-avoiding if all terminal subtrees have at least two branches, and achiral if all branches directly under any given node are equal. For example, the a(6) = 6 trees are 6, (111111), (222), ((11)(11)(11)), (33), ((111)(111)). - Gus Wiseman, Jul 13 2018. Updated Aug 22 2020.

From Gus Wiseman, Aug 20 2020: (Start)

Also the number of strict chains of divisors starting with n. For example, the a(n) chains for n = 1, 2, 4, 6, 8, 12 are:

1 2 4 6 8 12

2/1 4/1 6/1 8/1 12/1

4/2 6/2 8/2 12/2

4/2/1 6/3 8/4 12/3

6/2/1 8/2/1 12/4

6/3/1 8/4/1 12/6

8/4/2 12/2/1

8/4/2/1 12/3/1

12/4/1

12/4/2

12/6/1

12/6/2

12/6/3

12/4/2/1

12/6/2/1

12/6/3/1

(End)

a(n) is the number of chains including n of the divisor lattice of divisors of n, which is to say, a(n) is the number of (d_1,d_2,...,d_k) such that d_1 < d_2 < ... < d_k = n and d_i divides d_{i+1} for 1 <= i <= k-1. Using this definition, the recurrence a(n) = 1 + Sum_{0 < d < n, d|n} a(d) is evident by enumerating the preceding element of n in the chains. If we count instead the chains whose LCM of members is n, then a(1) would be 2 because the empty chain is included, and we would obtain 2*A074206(n). - Jianing Song, Aug 21 2024

A partition of n is complete if every number 1 to n can be represented as a sum of parts of the partition. This generalizes perfect partitions, where the representation for each number must be unique.

A partition is complete iff each part is no more than 1 more than the sum of all smaller parts. (This includes the smallest part, which thus must be 1.) - Franklin T. Adams-Watters, Mar 22 2007

For n > 0: a(n) = sum of n-th row in A261036 and also a(floor(n/2)) = A261036(n,floor((n+1)/2)). - Reinhard Zumkeller, Aug 08 2015

a(n+1) is the number of partitions of n such that each part is no more than 2 more than the sum of all smaller parts (generalizing Adams-Watters's criterion). Bijection: each partition counted by a(n+1) must contain a 1, removing that gives a desired partition of n. - Brian Hopkins, May 16 2017

A partition (x_1, ..., x_k) is complete if and only if 1, x_1, ..., x_k is a "regular sequence" (see A003513 for definition). As a result, the number of complete partitions with n parts is given by A003513(n+1). - Nathaniel Johnston, Jun 29 2023

For prime p, a(p)=3.

a(2^k) = 2^(k+1)-1.

For integers of the form n = p_1*p_2*...*p_k we have a(n) = A007047(k).

The value of a(n) depends only on the exponents in the prime factorization of n.

Inverse Moebius transform of A000041.

Row sums of triangle A137587. - Gary W. Adamson, Jan 27 2008

Row sums of triangle A168021. - Omar E. Pol, Nov 20 2009

Row sums of triangle A168017. Row sums of triangle A168018. - Omar E. Pol, Nov 25 2009

Sum of the partition numbers of the divisors of n. - Omar E. Pol, Feb 25 2014

Conjecture: for n > 6, a(n) is strictly increasing. - Franklin T. Adams-Watters, Apr 19 2014

Number of constant multiset partitions of multisets spanning an initial interval of positive integers with multiplicities an integer partition of n. - Gus Wiseman, Sep 16 2018