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

%I A350535 #14 Oct 23 2022 01:09:25
%S A350535 1,2,3,5,7,11,13,17,23,31,37,41,43,53,67,71,83,97,101,107,113,157,167,
%T A350535 181,191,193,223,233,251,283,317,347,373,421,431,487,521,563,577,613,
%U A350535 643,647,743,907,1033,1091,1103,1193,1201,1213,1277,1291,1423,1427,1471,1543,1583,1597
%N A350535 Integers that cannot be expressed as x + y + z + x*y*z for x,y,z >= 1.
%C A350535 Terms greater than 1 are prime.
%H A350535 Brian Conrey and Neil Shah, <a href="https://arxiv.org/abs/2112.15551">Which numbers are not the sum plus the product of three positive integers?</a>, arXiv:2112.15551 [math.NT], 2021.
%o A350535 (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
%o A350535 (Python)
%o A350535 from itertools import count, islice
%o A350535 def A350535_gen(startvalue=1): # generator of terms >= startvalue
%o A350535     for n in count(max(startvalue,1)):
%o A350535         flag = True
%o A350535         for x in range(1,n+1):
%o A350535             if 3*x+x**3 > n or not flag:
%o A350535                 break
%o A350535             for y in range(x,n+1):
%o A350535                 if x+2*y+x*y**2 > n:
%o A350535                     break
%o A350535                 if (n-x-y)%(1+x*y) == 0 and x+y*(2+x*y)<= n:
%o A350535                     flag = False
%o A350535                     break
%o A350535         if flag:
%o A350535             yield n
%o A350535 A350535_list = list(islice(A350535_gen(),30)) # _Chai Wah Wu_, Oct 21 2022
%Y A350535 Cf. A260803.
%K A350535 nonn
%O A350535 1,2
%A A350535 _Michel Marcus_, Jan 04 2022