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

%I A066755 #26 May 17 2025 05:56:06
%S A066755 1,2,4,6,10,14,16,20,24,26,34,36,40,44,46,50,54,56,60,66,70,74,76,84,
%T A066755 86,90,94,96,100,104,110,114,116,120,124,126,130,134,136,144,146,150,
%U A066755 156,160,164,170,176,180,184,186,190,194,196,204,206,210,214,220,224
%N A066755 Numbers m such that m^2 + 1 is not divisible by k^2 + 1 for any k in [1,m-1].
%C A066755 Equivalently, A066743(m)=1.
%C A066755 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, ...
%C A066755 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
%H A066755 Amiram Eldar, <a href="/A066755/b066755.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harry J. Smith)
%p A066755 a:= proc(n) option remember; local k; for k from 1+
%p A066755       `if`(n=1, 0, a(n-1)) while ormap(t->
%p A066755       irem(k^2+1, t)=0, [(j^2+1)$j=1..k-1]) do od; k
%p A066755     end:
%p A066755 seq(a(n), n=1..100);  # _Alois P. Heinz_, Sep 18 2019
%t A066755 a66743[ n_ ] := Length[ Select[ Range[ 1, n ], IntegerQ[ (n^2+1)/(#^2+1) ]& ] ]; Select[ Range[ 1, 300 ], a66743[ # ]==1& ]
%o A066755 (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
%Y A066755 Cf. A066743, A005574.
%K A066755 nonn
%O A066755 1,2
%A A066755 _Benoit Cloitre_, Jan 16 2002
%E A066755 Edited by _Dean Hickerson_, Jan 20 2002