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A249066
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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.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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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.
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MAPLE
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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:
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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