A391512 Primes expressible as a*b*c*d - (a+b+c+d) where a,b,c,d are distinct composite numbers.

2131, 3851, 4283, 4999, 5003, 6011, 6679, 6871, 7159, 7727, 7867, 7873, 8017, 8581, 8597, 8599, 9011, 9311, 9341, 9551, 9677, 9871, 10037, 10039, 10301, 10321, 10487, 10753, 11149, 11161, 11171, 11471, 11701, 11833, 12041, 12043, 12049, 12451, 12613, 12893, 12899, 12907, 12911, 13381, 13619

Offset

1,1

Comments

If a,b,c are distinct composites and d > 1 such that (a*b*c-1)*d and a+b+c are coprime, then by Dirichlet's theorem there are infinitely many terms of the form a*b*c*d*x - (a + b + c + d*x). For example, with a=4, b=6, c=9, d=2, the sequence includes all primes == 411 (mod 430).

Links

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

Example

a(4) = 4*6*10*21 - (4+6+10+21) = 4999.
The first term expressible in more than one way is a(16) = 6*8*12*15 - (6+8+12+15) = 6*9*10*16 - (6+9+10+16).

Maple

N:= 15000: # for terms <= N
R:= {}:
for i from 4 while i^4 - 4*i < N do
  if isprime(i) then next fi;
  for j from i+1 while i * j^3 - i - 3*j < N do
     if isprime(j) then next fi;
     for k from j+1 while i*j*k^2 - i-j-2*k < N do
       if isprime(k) then next fi;
       for l from k+1 do
         v:= i*j*k*l-(i+j+k+l);
         if v > N then break fi;
         if not isprime(l) and isprime(v) then R:= R union {v} fi
od od od od:
sort(convert(R, list));

Crossrefs

Cf. A391698, A391699.

Sequence in context: A210271 A066817 A110024 * A260068 A237070 A157768

Adjacent sequences: A391509 A391510 A391511 * A391513 A391514 A391515

Keyword

nonn

Author

Will Gosnell and Robert Israel, Dec 11 2025

Status

approved