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A247340 Numbers n such that each prime divisor of the semiprime n^2+1 is also a divisor of a^2+1 and b^2+1 respectively for some a, b < n.

%I A247340 #6 Sep 20 2014 14:20:14
%S A247340 3,8,30,46,50,76,100,144,254,266,274,286,334,380,456,494,504,516,520,
%T A247340 526,566,664,670,726,756,810,836,844,874,1040,1064,1086,1130,1164,
%U A247340 1216,1250,1300,1476,1714,1740,1800,1826,1834,1946,1950,2014,2194,2200,2220,2324
%N A247340 Numbers n such that each prime divisor of the semiprime n^2+1 is also a divisor of a^2+1 and b^2+1 respectively for some a, b < n.
%C A247340 Or numbers n such that n^2+1 = p*q, p and q primes => p | a^2+1 and q | b^2+1 for some a,b < n.
%C A247340 Subsequence of A085722 and except the first term, a(n) is even.
%C A247340 The squares of the sequence are 100, 144, 3364, 6084, 7396, 10404, 24964, 45796, 47524, 68644, 71824, 93636,...
%C A247340 Observation : a(n) = p*q => there exists a and b such that a^2+1 = m*p and b^2+1 = m*q. (see the examples).
%H A247340 Michel Lagneau, <a href="/A247340/b247340.txt">Table of n, a(n) for n = 1..1000</a>
%e A247340 3^2+1 = 2*5 => 1^1+1 = 2 and 2^2+1 = 5 ;
%e A247340 8^2+1 = 5*13 => 3^2+1 = 2*5 and 5^2+1 = 2*13 ;
%e A247340 30^2+1 = 17*53 => 13^2+1=2*5*17 and 23^2+1 = 2*5*53 ;
%e A247340 46^2+1 = 29*73 => 17^2+1 = 2*5*29 and 27^2+1=2*5*73 ;
%e A247340 50^2+1 = 41*61 => 9^2+1 = 2*41 and 11^2+1 = 2*61 ;
%e A247340 76^2+1 = 53*109 => 23^2+1 = 2*5*53 and 33^2+1 = 2*5*109 ;
%e A247340 100^2+1 = 73*137 => 27^2+1=2*5*73 and 37^2+1 = 2*5*137 ;
%e A247340 144^2+1 = 89*233 => 55^2+1 = 2*17*89 and 89^2+1 = 2*17*233 ;
%e A247340 254^2+1 = 149*433 => 105^2+1 = 2*37*149 and 179^2+1 = 2*37*433 ;
%e A247340 266^2+1 = 173*409 => 93^2+1 = 2*5^2*173 and 143^2+1 = 2*5^2*409.
%p A247340 with(numtheory):lst:={}:
%p A247340 for n from 1 to 3000 do:
%p A247340    x:=factorset(n^2+1):n0:=nops(x):
%p A247340      for i from 1 to n0 do:
%p A247340       lst:=lst union {x[i]}:
%p A247340      od:
%p A247340       lst1:={}:nn:=n+1:xx:=factorset(nn^2+1):nn0:=nops(xx):
%p A247340         for j from 1 to nn0 do:
%p A247340          lst1:=lst1 union {xx[j]}:
%p A247340         od:
%p A247340         if
%p A247340          nn0=2
%p A247340          and bigomega(nn^2+1)=2
%p A247340          and {xx[1],xx[2]} intersect lst = {xx[1],xx[2]}
%p A247340          then
%p A247340          printf(`%d, `,n+1):
%p A247340          else
%p A247340         fi:
%p A247340         lst:=lst union lst1:
%p A247340   od:
%Y A247340 Cf. A085722, A144255.
%K A247340 nonn
%O A247340 1,1
%A A247340 _Michel Lagneau_, Sep 14 2014