cp's OEIS Frontend

Except rows 3, 4, 10, 11, 21 of triangle A008289, all the other rows up to row number 10^6 contain a single maximal value.

Conjecture: for n >= 22 there is a unique maximal value in row n of triangle A008289.

Compositions into distinct parts are equivalent to (1,1)-avoiding compositions. - Gus Wiseman, Jun 25 2020

All terms are odd. - Alois P. Heinz, Apr 09 2021

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

Conjecture: Reverse the rows of the table to get an infinite lower-triangular matrix b with 1's on the main diagonal. The third diagonal of the inverse of b is minus A137719. - George Beck, Oct 26 2019

Proof: The reversed rows yield the matrix I+N where N is strictly lower triangular, N[i,j] = 0 for j >= i, having its 2nd diagonal equal to the 2nd column (1, 0, 1, 0, 1, ...): N[i+1,i] = A000035(i), i >= 1, and 3rd diagonal equal to the 3rd column of this triangle, (2, 1, 2, 3, 3, 3, ...): N[i+2,i] = A137719(i), i >= 1. It is known that (I+N)^-1 = 1 - N + N^2 - N^3 +- .... Here N^2 has not only the second but also the 3rd diagonal zero, because N^2[i+2,i] = N[i+2,i+1]*N[i+1,i] = A000035(i+1)*A000035(i) = 0. Therefore the 3rd diagonal of (I+N)^-1 is equal to -A137719 without leading 0. - M. F. Hasler, Oct 27 2019

From Gus Wiseman, Aug 27 2023: (Start)

Also the number of ways to write n-k as a nonnegative linear combination of a strict integer partition of k. Also the number of ways to write n as a (strictly) positive linear combination of a strict integer partition of k. Row n=7 counts the following:

7*1 . 1*2+5*1 1*3+4*1 1*3+2*2 1*5+2*1 1*7

2*2+3*1 2*3+1*1 1*4+3*1 1*3+1*2+2*1 1*4+1*3

3*2+1*1 1*5+1*2

1*6+1*1

1*4+1*2+1*1

(End)

Regarded as a triangular table, this is another version of the number of partitions of n into k parts, A008284. - Franklin T. Adams-Watters, Dec 18 2006

From Gus Wiseman, Feb 10 2021: (Start)

T(n,k) is also the number of partitions of n with greatest part k, if we assume the greatest part of an empty partition to be 0. Row n = 9 counts the following partitions:

111111111 22221 333 432 54 63 72 81 9

222111 3222 441 522 621 711

2211111 3321 4221 531 6111

21111111 32211 4311 5211

33111 42111 51111

321111 411111

3111111

(End)

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Ramanujan theta functions: phi(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

A way of writing n as a (presumed nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).