

A236998


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


8



0, 0, 1, 0, 0, 1, 2, 0, 2, 2, 1, 1, 2, 1, 1, 1, 1, 3, 3, 2, 2, 4, 3, 1, 3, 1, 3, 1, 1, 2, 2, 1, 4, 4, 3, 3, 1, 1, 5, 2, 3, 7, 2, 5, 3, 4, 3, 2, 7, 3, 2, 3, 4, 6, 2, 1, 7, 5, 3, 2, 2, 4, 4, 2, 6, 4, 3, 5, 5, 7, 4, 3, 2, 6, 4, 2, 7, 5, 5, 4, 4, 2, 4, 8, 2, 7, 5, 7, 3, 3, 8, 6, 7, 5, 7, 3, 9, 3, 7, 5
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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

ZhiWei Sun, Table of n, a(n) for n = 1..10000


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

Cf. A000010, A000290, A234246, A236567, A237016.
Sequence in context: A213013 A242667 A059581 * A297116 A163542 A061895
Adjacent sequences: A236995 A236996 A236997 * A236999 A237000 A237001


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Feb 02 2014


STATUS

approved



