A391699 Primes expressible as a*b*c - (a+b+c) where a, b and c are distinct composite numbers.

197, 337, 409, 547, 571, 617, 691, 727, 757, 827, 887, 967, 971, 977, 1039, 1049, 1117, 1151, 1163, 1223, 1279, 1301, 1303, 1439, 1459, 1583, 1597, 1667, 1693, 1723, 1741, 1753, 1759, 1787, 1801, 1877, 1907, 1931, 1951, 1973, 1987, 1999, 2003, 2017, 2083, 2087, 2099, 2113, 2129, 2131, 2221, 2267

Offset

1,1

Comments

For any distinct composites a,b and any d>1 such that (a*b-1)*d and a+b are coprime, by Dirichlet's theorem there are infinitely many primes a*b*d*k - (a + b + d*k). For example, taking a=4, b=6 and d=3, the sequence includes the infinitely many primes == 59 (mod 69) except 59.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

a(3) = 409 is a term because 409 = 6 * 8 * 9 - (6 + 8 + 9) is prime, where 6, 8, 9 are distinct composites.

Maple

N:= 10^4: # for terms <= N
R:= {}:
for i from 4 while i^3 - 3*i < N do
  if isprime(i) then next fi;
  for j from i+1 while i * j^2 - i - 2*j < N do
     if isprime(j) then next fi;
     for k from j+1 do
         v:= i*j*k-(i+j+k);
         if v > N then break fi;
         if not isprime(k) and isprime(v) then R:= R union {v} fi
od od od:
R:=sort(convert(R, list));

Crossrefs

Cf. A391512, A391698.

Sequence in context: A142274 A142624 A294187 * A158223 A142503 A142568

Adjacent sequences: A391696 A391697 A391698 * A391700 A391701 A391702

Keyword

nonn

Author

Robert Israel, Dec 17 2025

Status

approved