A328525 Numbers k such that (k-1)*k*(k+1) = (k-1)*(1+u) = k*(1+v) = (k+1)*(1+w) with primes u, v, w.

3, 5, 9, 21, 55, 131, 145, 155, 231, 259, 265, 449, 495, 561, 595, 1045, 1051, 1365, 1409, 1491, 1549, 1849, 1989, 2001, 2101, 2469, 2785, 3365, 3621, 3641, 3669, 3845, 3911, 4285, 4951, 5181, 5465, 6049, 6699, 7189, 7229, 8219, 8629, 9175, 9521, 9539, 9631, 9729

Offset: 1

Author

Frank Ellermann, Feb 24 2020

Keywords

• nonn

Examples

			3 is a term because 2*3*4 = 2*(1+11) = 3*(1+7) = 4*(1+5) with primes 11, 7, 5.
9 is a term because 8*9*10 = 8*(1+89) = 9*(1+79) = 10*(1+71) with primes 89, 79, 71.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040.

Intersection of A002328, A028870 and A045546.

Programs

Maple

q:= k-> andmap(isprime, (t-> [t-1, t-k, t+k])(k^2-1)):
select(q, [$1..10000])[];  # Alois P. Heinz, Feb 25 2020

Mathematica

Select[Range[2, 10^4], AllTrue[{(# - 1)*#, #*(# + 1), (# + 1)*(# - 1)} - 1, PrimeQ] &] (* Amiram Eldar, Feb 24 2020 *)

PARI

isok(k) = isprime(k*(k+1)-1) && isprime((k+1)*(k-1)-1) && isprime(k*(k-1)-1); \\ Michel Marcus, Feb 25 2020

Rexx

S = 3
do N = 5 to 595 by 2
   if NOPRIME( N*N +N -1 ) then  iterate N
   if NOPRIME( N*N    -2 ) then  iterate N
   if NOPRIME( N*N -N -1 ) then  iterate N
   S = S || ',' N
end N
say S

Extensions

More terms from Amiram Eldar, Feb 24 2020