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

0,7

Links

Table of n, a(n) for n=0..94.

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.

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

Example

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

Prog

(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

Crossrefs

Cf. A350535, A357809.

Sequence in context: A318050 A053604 A066958 * A066851 A288571 A270028

Adjacent sequences: A357059 A357060 A357061 * A357063 A357064 A357065

Keyword

nonn

Author

Charles R Greathouse IV, Oct 13 2022

Status

approved