A350535 Integers that cannot be expressed as x + y + z + x*y*z for x,y,z >= 1.

1, 2, 3, 5, 7, 11, 13, 17, 23, 31, 37, 41, 43, 53, 67, 71, 83, 97, 101, 107, 113, 157, 167, 181, 191, 193, 223, 233, 251, 283, 317, 347, 373, 421, 431, 487, 521, 563, 577, 613, 643, 647, 743, 907, 1033, 1091, 1103, 1193, 1201, 1213, 1277, 1291, 1423, 1427, 1471, 1543, 1583, 1597

Offset: 1

Author

Michel Marcus, Jan 04 2022

Keywords

• nonn

Comments

Terms greater than 1 are prime.

Links

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

Crossrefs

Cf. A260803.

Programs

PARI

isok(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))) == 0; \\ see A260803

Python

from itertools import count, islice
def A350535_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        flag = True
        for x in range(1,n+1):
            if 3*x+x**3 > n or not flag:
                break
            for y in range(x,n+1):
                if x+2*y+x*y**2 > n:
                    break
                if (n-x-y)%(1+x*y) == 0 and x+y*(2+x*y)<= n:
                    flag = False
                    break
        if flag:
            yield n
A350535_list = list(islice(A350535_gen(),30)) # Chai Wah Wu, Oct 21 2022