A357062 Number of ordered solutions to n = x*y*z + x + y + z in positive integers.

0, 0, 0, 0, 1, 0, 3, 0, 3, 3, 3, 0, 9, 0, 4, 6, 6, 0, 9, 3, 9, 6, 3, 0, 18, 3, 6, 6, 9, 3, 15, 0, 9, 12, 6, 6, 19, 0, 3, 9, 21, 0, 18, 0, 12, 12, 6, 6, 21, 6, 9, 12, 9, 0, 24, 6, 18, 6, 3, 6, 33, 6, 6, 12, 15, 6, 18, 0, 15, 15, 15, 0, 33, 6, 6, 18, 13, 6, 21, 3, 21, 9, 9, 0, 36, 12, 9, 12, 18, 9, 27, 6, 9, 9, 6

Offset: 0

Author

Charles R Greathouse IV, Oct 13 2022

Keywords

• nonn

Examples

			6 = 2*1*1 + 2 + 1 + 1 = 1*2*1 + 1 + 2 + 1 = 1*1*2 + 1 + 1 + 2, so a(6) = 3.
		

Links

• Brian Conrey and Neil Shah, Which numbers are not the sum plus the product of three positive integers?, 2021 preprint. arXiv:2112.15551 [math.NT], 2021-2022.

Crossrefs

Cf. A350535, A357809.

Programs

PARI

a(n)=sum(x=1,(n-1)\2, my(s); for(y=1,x, my(m=x*y+1); if(m+x+y>n, break); my(N=n-y-x,z); if(N%m, next); z=N/m; z<=y && s += [1,3,6][#Set([x,y,z])]); s)

Python

from sympy.utilities.iterables import combinations_with_replacement
from math import prod
def A357062(n): return sum(max(1,3*(len(set(d))-1)) for d in combinations_with_replacement(range(1,n+1),3) if prod(d)+sum(d) == n) # Chai Wah Wu, Oct 21 2022

Formula

Conrey & Shah prove that a(n) << n^(1.3) * log n * (log log n)^4, and conjecture that a(n) << n^e for any e > 0.

Conrey & Shah prove that the average value of a(n) is (log n)^2/2, in the sense that Sum_{k <= n} a(k) ~ n*(log n)^2/2.

a(n) = 0 iff n = 0 or n belongs to A350535. - Rémy Sigrist, Oct 21 2022