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.

3, 8, 30, 46, 50, 76, 100, 144, 254, 266, 274, 286, 334, 380, 456, 494, 504, 516, 520, 526, 566, 664, 670, 726, 756, 810, 836, 844, 874, 1040, 1064, 1086, 1130, 1164, 1216, 1250, 1300, 1476, 1714, 1740, 1800, 1826, 1834, 1946, 1950, 2014, 2194, 2200, 2220, 2324

Offset: 1

Author

Michel Lagneau, Sep 14 2014

Keywords

• nonn

Comments

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.

Subsequence of A085722 and except the first term, a(n) is even.

The squares of the sequence are 100, 144, 3364, 6084, 7396, 10404, 24964, 45796, 47524, 68644, 71824, 93636,...

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).

Examples

			3^2+1 = 2*5 => 1^1+1 = 2 and 2^2+1 = 5 ;
8^2+1 = 5*13 => 3^2+1 = 2*5 and 5^2+1 = 2*13 ;
30^2+1 = 17*53 => 13^2+1=2*5*17 and 23^2+1 = 2*5*53 ;
46^2+1 = 29*73 => 17^2+1 = 2*5*29 and 27^2+1=2*5*73 ;
50^2+1 = 41*61 => 9^2+1 = 2*41 and 11^2+1 = 2*61 ;
76^2+1 = 53*109 => 23^2+1 = 2*5*53 and 33^2+1 = 2*5*109 ;
100^2+1 = 73*137 => 27^2+1=2*5*73 and 37^2+1 = 2*5*137 ;
144^2+1 = 89*233 => 55^2+1 = 2*17*89 and 89^2+1 = 2*17*233 ;
254^2+1 = 149*433 => 105^2+1 = 2*37*149 and 179^2+1 = 2*37*433 ;
266^2+1 = 173*409 => 93^2+1 = 2*5^2*173 and 143^2+1 = 2*5^2*409.
		

Links

• Michel Lagneau, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A085722, A144255.

Programs

Maple

with(numtheory):lst:={}:
for n from 1 to 3000 do:
   x:=factorset(n^2+1):n0:=nops(x):
     for i from 1 to n0 do:
      lst:=lst union {x[i]}:
     od:
      lst1:={}:nn:=n+1:xx:=factorset(nn^2+1):nn0:=nops(xx):
        for j from 1 to nn0 do:
         lst1:=lst1 union {xx[j]}:
        od:
        if
         nn0=2
         and bigomega(nn^2+1)=2
         and {xx[1],xx[2]} intersect lst = {xx[1],xx[2]}
         then
         printf(`%d, `,n+1):
         else
        fi:
        lst:=lst union lst1:
  od: