A230486 Numbers n such that n^n is representable as the sum of two nonzero squares.

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

Offset: 1

Author

Alex Ratushnyak, Oct 20 2013

Keywords

• nonn

Comments

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

Examples

			5^5 = 55^2 + 10^2.
10^10 = 99712^2 + 7584^2.
13^13 = 17106843^2 + 3198598^2.
17^17 = 28735037644^2 + 1240110271^2.
		

References

• G. H. Hardy and E. M. Wright, Theory of Numbers, Oxford, Sixth Edition, 2008, p. 395.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000312 (n^n), A004431, A132777.

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.

Programs

Mathematica

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 *)

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)))
A230486_list = list(islice(A230486_gen(),20)) # Chai Wah Wu, May 15 2023

Formula

A230486 = { n | A000312(n) is in A000404 } = A004277*A002144 U A008846. - M. F. Hasler, Sep 02 2018

Extensions

Extended by T. D. Noe, Oct 21 2013