A242109 - OEIS

%I #10 Aug 21 2014 22:49:15

%S 2,2917,13457,15377,15877,21317,78401,147457,190097,215297,217157,

%T 287297,401957,414737,577601,1299601,1308737,1313317,1378277,1547537,

%U 1623077,1664101,1731857,1742401,1822501,1887877,1976837,2044901,2390117,2421137,2446097,2483777

%N First of two consecutive (primes of the form n^2+1) with no semiprime of the same form between them.

%e 2 is in the sequence because there is no semiprime between the two primes 1^2 + 1 = 2 and 2^2 + 1 = 5 of the form k^2 + 1.

%e 2917 is in the sequence because there is no semiprime between the two primes 54^2 + 1 = 2917 and 56^2 + 1 = 3127 : 55^2 + 1 = 3026 = 2*17*89 is not a semiprime.

%p with(numtheory):nn:=2000: lst:={}:

%p for n from 1 to nn do:

%p   if type(n^2+1,prime)=true

%p     then

%p     lst:=lst union {n}:

%p     else

%p   fi:

%p od:

%p n1:=nops(lst):

%p   for m from 1 to n1-1 do:

%p     i1:=lst[m]:i2:=lst[m+1]:ii:=0:

%p      for k from i1+1 to i2-1 do:

%p        x:=k^2+1:y:=factorset(x):

%p          if bigomega(x)=2 and nops(y)=2

%p          then

%p          ii:=ii+1:

%p          else

%p        fi:

%p     od:

%p     if ii=0

%p     then

%p     printf(`%d, `,i1^2+1):

%p     else

%p     fi:

%p od:

%o (PARI)

%o for(n=1,10^4,if(isprime(n^2+1),k=1;while(!isprime((n+k)^2+1),k++);c=0;for(i=1,k-1,d=factor((n+i)^2+1);s=sum(j=1,#d[,1],d[j,2]);if(s==2,c++;break));if(c==0,print1(n^2+1,", ")))) \\ _Derek Orr_, Aug 15 2014

%Y Cf. A002496, A005574.

%K nonn

%O 1,1

%A _Michel Lagneau_, Aug 15 2014