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

1,2

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).

Links

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

Example

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.

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:

Crossrefs

Cf. A002522, A002496, A234759, A238138.

Sequence in context: A366719 A129733 A084160 * A268722 A213828 A324003

Adjacent sequences: A238136 A238137 A238138 * A238140 A238141 A238142

Keyword

nonn

Author

Michel Lagneau, Feb 18 2014

Status

approved