A306723 Composite squarefree numbers k such that k^2+1 is divisible by p-1, where p are all the prime factors of k.

33, 36003, 426747, 220067817

Offset: 1

Author

Paolo P. Lava, Mar 06 2019

Keywords

• nonn
• hard
• more

Comments

Tested up to 3*10^10. - Giovanni Resta, Mar 06 2019

Examples

			Prime factors of 33 are 3, 11 and 33^2+1 = 1090, 1090/2 = 545, 1090/10 = 109.
Prime factors of 220067817 are 3, 59, 131, 9491 and 220067817^2+1 = 48429844079145490, 48429844079145490/2 = 24214922039572745, 48429844079145490/58 = 834997311709405, 48429844079145490/130 = 372537262147273, 48429844079145490/9490 = 5103250166401.
		

Crossrefs

Cf. A002997, A006972, A208728, A225711, A304291, A306685.

Programs

Maple

with(numtheory): P:=proc(q) local a,k,ok,n;
for n from 1 to q do if not isprime(n) and issqrfree(n) then a:=factorset(n); ok:=1; for k from 1 to nops(a) do if frac((n^2+1)/(a[k]+1))>0 then ok:=0; break; fi; od; if ok=1 then print(n); fi; fi; od; end: P(10^20);

PARI

isok(n) = {if (issquarefree(n) && !isprime(n) && (n>1), my(f = factor(n)[,1], x = n^2+1); for (k=1, #f, if ((x % (f[k]-1)), return (0));); return (1);); return (0);} \\ Michel Marcus, Mar 12 2019

Extensions

a(4) from Giovanni Resta, Mar 06 2019