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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.

%I A328525 #26 Sep 06 2023 13:41:15
%S A328525 3,5,9,21,55,131,145,155,231,259,265,449,495,561,595,1045,1051,1365,
%T A328525 1409,1491,1549,1849,1989,2001,2101,2469,2785,3365,3621,3641,3669,
%U A328525 3845,3911,4285,4951,5181,5465,6049,6699,7189,7229,8219,8629,9175,9521,9539,9631,9729
%N 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.
%H A328525 Alois P. Heinz, <a href="/A328525/b328525.txt">Table of n, a(n) for n = 1..10000</a>
%e A328525 3 is a term because 2*3*4 = 2*(1+11) = 3*(1+7) = 4*(1+5) with primes 11, 7, 5.
%e A328525 9 is a term because 8*9*10 = 8*(1+89) = 9*(1+79) = 10*(1+71) with primes 89, 79, 71.
%p A328525 q:= k-> andmap(isprime, (t-> [t-1, t-k, t+k])(k^2-1)):
%p A328525 select(q, [$1..10000])[];  # _Alois P. Heinz_, Feb 25 2020
%t A328525 Select[Range[2, 10^4], AllTrue[{(# - 1)*#, #*(# + 1), (# + 1)*(# - 1)} - 1, PrimeQ] &] (* _Amiram Eldar_, Feb 24 2020 *)
%o A328525 (Rexx)
%o A328525 S = 3
%o A328525 do N = 5 to 595 by 2
%o A328525    if NOPRIME( N*N +N -1 ) then  iterate N
%o A328525    if NOPRIME( N*N    -2 ) then  iterate N
%o A328525    if NOPRIME( N*N -N -1 ) then  iterate N
%o A328525    S = S || ',' N
%o A328525 end N
%o A328525 say S
%o A328525 (PARI) isok(k) = isprime(k*(k+1)-1) && isprime((k+1)*(k-1)-1) && isprime(k*(k-1)-1); \\ _Michel Marcus_, Feb 25 2020
%Y A328525 Cf. A000040.
%Y A328525 Intersection of A002328, A028870 and A045546.
%K A328525 nonn
%O A328525 1,1
%A A328525 _Frank Ellermann_, Feb 24 2020
%E A328525 More terms from _Amiram Eldar_, Feb 24 2020