A101423 Number of different cuboids with volume p^3 * q^n, where p,q are distinct prime numbers.

3, 6, 12, 19, 28, 38, 51, 64, 80, 97, 116, 136, 159, 182, 208, 235, 264, 294, 327, 360, 396, 433, 472, 512, 555, 598, 644, 691, 740, 790, 843, 896, 952, 1009, 1068, 1128, 1191, 1254, 1320, 1387, 1456, 1526, 1599, 1672, 1748, 1825, 1904, 1984, 2067, 2150, 2236

Offset

0,1

Comments

Subsequence of A034836, which gives the number of cuboids for volume n.

Links

Table of n, a(n) for n=0..50.

Formula

a(n) = A034836(2^3*3^n) = A034836(3^3*2^n) = A034836(p^3*q^n) for p,q distinct primes.

Empirical g.f.: -(x^3+3*x^2+3*x+3)/((x+1)*(x^2+x+1)*(x-1)^3). - Colin Barker, Mar 28 2014

If 3|n, then a(n) = (5*(n+2)*(n+1)+2)/6 + floor(n/2) + 1. Otherwise a(n) = (5*(n+2)*(n+1))/6 + floor(n/2) + 1. - Chai Wah Wu, May 28 2026

Prog

(Python)
def A101423(n): return (5*(n+2)*(n+1)+(0 if n%3 else 2))//6+(n>>1)+1 # Chai Wah Wu, May 28 2026

Crossrefs

Cf. A034836, A101424, A101425, A101426, A101427.

Sequence in context: A345868 A318872 A267591 * A263511 A180622 A125851

Adjacent sequences: A101420 A101421 A101422 * A101424 A101425 A101426

Keyword

nonn,changed

Author

Anthony C Robin, Jan 17 2005

Extensions

Corrected, edited and extended by Ray Chandler, Dec 19 2008

a(0)=3 prepended and g.f. edited by Alois P. Heinz, Oct 05 2016

Status

approved