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A230486
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Numbers n such that n^n is representable as the sum of two nonzero squares.
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2
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5, 10, 13, 17, 20, 25, 26, 29, 30, 34, 37, 40, 41, 50, 52, 53, 58, 60, 61, 65, 68, 70, 73, 74, 78, 80, 82, 85, 89, 90, 97, 100, 101, 102, 104, 106, 109, 110, 113, 116, 120, 122, 125, 130, 136, 137, 140, 145, 146, 148, 149, 150, 156, 157, 160, 164, 169, 170
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OFFSET
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1,1
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COMMENTS
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If n is even, then n must have a prime factor of the form 4k+1. If n is odd, then all prime factors must be of the form 4k+1. - T. D. Noe, Oct 21 2013
The above is also a sufficient condition: the sequence consists exactly in even multiples of Pythagorean primes A002144, and products of such primes (A008846). - M. F. Hasler, Sep 02 2018
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REFERENCES
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G. H. Hardy and E. M. Wright, Theory of Numbers, Oxford, Sixth Edition, 2008, p. 395.
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LINKS
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FORMULA
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EXAMPLE
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5^5 = 55^2 + 10^2.
10^10 = 99712^2 + 7584^2.
13^13 = 17106843^2 + 3198598^2.
17^17 = 28735037644^2 + 1240110271^2.
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MATHEMATICA
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t = {}; Do[f = FactorInteger[n]; p = Transpose[f][[1]]; If[EvenQ[n], If[MemberQ[Mod[p, 4], 1], AppendTo[t, n]], If[Union[Mod[p, 4]] == {1}, AppendTo[t, n]]], {n, 2, 200}]; t (* T. D. Noe, Oct 21 2013 *)
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PROG
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(PARI) select( is_A230486(n)={(n=factor(n)[, 1]%4) && if(n[1]==2, Set(n)[1]==1, Set(n)==[1])}, [1..200]) \\ M. F. Hasler, Sep 02 2018
(Python)
from itertools import count, islice
from sympy import primefactors
def A230486_gen(startvalue=2): # generator of terms >= startvalue
return filter(lambda n:all(p&3==1 for p in primefactors(n)) if n&1 else any(p&3==1 for p in primefactors(n)), count(max(startvalue, 2)))
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CROSSREFS
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A subsequence of A000404 (numbers that are the sum of 2 nonzero squares).
Sequence A002144 (primes of the form 4k + 1) and A008846 (products of such primes) are subsequences.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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