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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.
3
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
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
KEYWORD
nonn
AUTHOR
Michel Lagneau, Feb 18 2014
STATUS
approved