

A236998


a(n) = {0 < k < n/2: phi(k)*phi(nk) is a square}, where phi(.) is Euler's totient function.


8



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OFFSET

1,7


COMMENTS

Conjecture: (i) a(n) > 0 for all n > 8.
(ii) If n > 20, then phi(k)*phi(nk) + 1 is a square for some 0 < k < n/2.
(iii) If n > 1 is not among 4, 7, 60, 199, 267, then k*phi(nk) is a square for some 0 < k < n.
We have verified part (i) of the conjecture for n up to 2*10^6.


LINKS



EXAMPLE

a(17) = 1 since phi(5)*phi(12) = 4*4 = 4^2.
a(24) = 1 since phi(4)*phi(20) = 2*8 = 4^2.
a(56) = 1 since phi(8)*phi(48) = 4*16 = 8^2.


MATHEMATICA

SQ[n_]:=IntegerQ[Sqrt[n]]
p[n_, k_]:=SQ[EulerPhi[k]*EulerPhi[nk]]
a[n_]:=Sum[If[p[n, k], 1, 0], {k, 1, (n1)/2}]
Table[a[n], {n, 1, 100}]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



