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
Keywords
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.
Programs
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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)
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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