A249066 a(n) is the number of new prime distinct divisors of n^2+1 not already present in m^2+1 for all m < n.

1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1

Offset

1,1

Comments

a(n) = 0 or 1. Proof:

If n^2+1 is composite, it is always possible to write n^2+1 = x*y where x and y are two integers. Suppose a(n)=2 with x>n and y>n (if x<n or y<n, then x or y is divisor of (n-x)^2+1 or (n-y)^2+1). So, x*y > n^2+1, a contradiction.

Links

Michel Lagneau, Table of n, a(n) for n = 1..10000

Formula

a(A002313(n)) = 0.

a(A005574(n)) = 1.

Example

a(3)=0 because A002522(3)= 2*5 and the prime divisors 2 and 5 exist already with A002522(1)= 2 and A002522(2)= 5.
a(4)= 1 because A002522(4)=17 is prime.
a(5)= 1 because A002522(5)=2*13 and the prime divisor 13 appears for the first time.

Maple

with(numtheory): nn:=600:lst:={0}:for n from 1 to 100 do:x:=factorset(n^2+1):lst1:=lst intersect x:n0:=nops(x minus lst1): printf(`%d, `, n0):lst:=lst union x:od:

Crossrefs

Cf. A002313, A002522, A005574.

Sequence in context: A051023 A247795 A030657 * A176178 A267778 A285384

Adjacent sequences: A249063 A249064 A249065 * A249067 A249068 A249069

Keyword

nonn

Author

Michel Lagneau, Oct 20 2014

Status

approved