|
%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
|
