A260803 Number of ways to write n as n = x*y*z + x + y + z where 1 <= x <= y <= z <= n.

0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 2, 1, 2, 0, 2, 1, 2, 1, 1, 0, 4, 1, 2, 1, 2, 1, 3, 0, 2, 2, 2, 1, 5, 0, 1, 2, 4, 0, 4, 0, 3, 2, 2, 1, 4, 2, 2, 2, 2, 0, 5, 1, 4, 1, 1, 2, 6, 1, 2, 2, 4, 1, 4, 0, 3, 3, 3, 0, 6, 1, 2, 3, 4, 1, 4, 1, 4, 2, 2, 0, 7, 2, 3, 2, 4, 2, 5, 1, 2, 2, 2, 2, 8, 0, 3, 4

Offset: 0

Author

David A. Corneth, Jul 31 2015

Keywords

• nonn

Comments

From Vladimir Shevelev, Aug 02 2015: (Start)

Conjecture: liminf_{n->infinity} (a(n)) = 0.

Note that this limit should realize on a subsequence of primes. Indeed, if n>=4 is a composite number, then n = p*q, p>=2, q>=2. If p <= q, then, for x=1, y = p-1, z = q-1, we have x*y*z + x + y + z = 1*(p-1)*(q-1) + 1 + (p-1) + (q-1) = p*q = n; so a(n) >= 1. If p > q, then we set x=1, y = q-1, z = p-1, and again a(n) >= 1.

Note also that primes r for which a(r) = 0 should grow fast enough. Indeed, r should not be a prime of the form (2*t+1)*k + t + 2, 2 <= t <= k, (*) where t==0 or 2 (mod 3).

Indeed, in this case r = x*y*z + x + y + z for x = 2, y = t, z = k. Since gcd(2*t+1, t+2) = gcd(2*(t+2)-3, t+2)=1, then for every considered t and k>=t, the progression (*) contains infinitely many primes r for which a(r) >= 1.

Finally, note that limsup_{n->infinity} (a(n)) = infinity. Indeed, this limit is realized, say, on primorials (A002110), since, when m goes to infinity, the number of representations of A002110(m) of the form p*q tends to infinity. So on primorials >1 we have a strictly monotonic subsequence: 0,1,3,8,25,46,78,164 ... (the terms 46 and 78 were calculated by Michel Marcus, 164 - by David A. Corneth). (End)

Links

• David A. Corneth, Table of n, a(n) for n = 0..9999
• Brian Conrey and Neil Shah, Which numbers are not the sum plus the product of three positive integers?, arXiv:2112.15551 [math.NT], 2021.
• Vladimir Shevelev, Representation of positive integers by the form x1...xk+x1+...+xk, arXiv:1508.03970 [math.NT], 2015.

Crossrefs

Cf. A071693, A008619, A260804.

Programs

Mathematica

a[n_] := Sum[Sum[Boole[Mod[n-x-y, x y + 1] == 0 && n-x >= y(x y + 2)], {y, x, (n - x(1+x^2))/2 // Floor}], {x, 1, n/3 // Floor}];
Table[a[n], {n, 0, 99}] (* Jean-François Alcover, Sep 20 2018, after M. F. Hasler *)

PARI

a(n)=sum(x=1,n\3,sum(y=x,(n-x*(1+x^2))\2,(n-x-y)%(x*y+1)==0&&n-x>=y*(x*y+2))) \\ M. F. Hasler, Jul 31 2015

Formula

a(n) = A071693(n) - A008619(n) = A071693(n) - floor(n/2) - 1.