A066755 Numbers m such that m^2 + 1 is not divisible by k^2 + 1 for any k in [1,m-1].

1, 2, 4, 6, 10, 14, 16, 20, 24, 26, 34, 36, 40, 44, 46, 50, 54, 56, 60, 66, 70, 74, 76, 84, 86, 90, 94, 96, 100, 104, 110, 114, 116, 120, 124, 126, 130, 134, 136, 144, 146, 150, 156, 160, 164, 170, 176, 180, 184, 186, 190, 194, 196, 204, 206, 210, 214, 220, 224

Offset: 1

Author

Benoit Cloitre, Jan 16 2002

Keywords

• nonn

Comments

Equivalently, A066743(m)=1.

If m^2 + 1 is prime, m is in the sequence; i.e., the sequence contains A005574. But so are many other values of m: 34, 44, 46, 50, 60, 70, 76, 86, 96, ...

The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are , 5, 29, 247, 2354, 23329, 232646, 2324131, ... . Apparently, the asymptotic density of this sequence exists and equals 0.232... . - Amiram Eldar, May 17 2025

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Crossrefs

Cf. A066743, A005574.

Programs

Maple

a:= proc(n) option remember; local k; for k from 1+
      `if`(n=1, 0, a(n-1)) while ormap(t->
      irem(k^2+1, t)=0, [(j^2+1)$j=1..k-1]) do od; k
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 18 2019

Mathematica

a66743[ n_ ] := Length[ Select[ Range[ 1, n ], IntegerQ[ (n^2+1)/(#^2+1) ]& ] ]; Select[ Range[ 1, 300 ], a66743[ # ]==1& ]

PARI

{ n=0; for (m=1, 10^10, k=1; b=1; t=m^2 + 1; while (k < m - 1, if (t%(k^2 + 1)==0, b=0; break); k++); if (b, write("b066755.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Mar 23 2010

Extensions

Edited by Dean Hickerson, Jan 20 2002