A238139 a(n) is the smallest prime divisor (not yet in the sequence) of all composite numbers of the form m^2+1 between the primes A002496(n) and A002496(n+1), or 0 if there is no such prime.

0, 2, 13, 5, 17, 113, 29, 53, 313, 37, 137, 41, 89, 241, 61, 97, 233, 101, 73, 193, 557, 229, 601, 157, 8581, 109, 337, 293, 4993, 181, 14621, 433, 197, 149, 21013, 509, 277, 281, 521, 11329, 257, 173, 1321, 6917, 373, 389, 3037, 821, 7109, 353, 773, 397, 457

Offset: 1

Author

Michel Lagneau, Feb 18 2014

Keywords

• nonn

Comments

By convention, a(1) = 0 because there are no composite number of the form m^2+1 between A002496(1)=2 and A002496(2)=5.

a(n) = 0 when all divisors of the numbers of the form m^2+1 between the primes A002496(n) and A002496(n+1) already exist in the sequence.

Note that a(n) = 0 for n = 1, 62, 149, 257, 281, 286,...(see A238138).

Examples

			a(7) = 29 because the composites of the form m^2+1 between the two primes A002496(7)= 16^2+1 = 257 and A002496(8)= 20^2+1=401 are:
17^2+1= 2*5*29;
18^2+1 = 5*5*13;
19^2+1=2*181 and the smallest prime divisor not yet in the sequence is 29 because 2, 5 and 13 are already in the sequence.
		

Links

• Michel Lagneau, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A002522, A002496, A234759, A238138.

Programs

Maple

with(numtheory):lst:={}: lst2:={}:T:=array(1..2000):kk:=1:k:=0:for n from 2 by 2 to 500 do: p:=n^2+1:if type(p, prime)=true then k:=k+1:T[k]:=p:else fi:od:for i from 1 to k-1 do:lst1:={}:a:=sqrt(T[i]-1):b:=sqrt(T[i+1]-1):for j from a+1 to b-1 do:y:=factorset(j^2+1):lst1:=lst1 union y:od:lst1:=lst1 minus lst: if lst1<>{} then kk:=kk+1: printf(`%d, `,lst1[1]):lst:=lst union {lst1[1]}:else kk:=kk+1: printf(`%d, `,0):fi:od: