login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A236998 a(n) = |{0 < k < n/2: phi(k)*phi(n-k) 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

Conjecture: (i) a(n) > 0 for all n > 8.

(ii) If n > 20, then phi(k)*phi(n-k) + 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(n-k) is a square for some 0 < k < n.

We have verified part (i) of the conjecture for n up to 2*10^6.

LINKS

Zhi-Wei 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[n-k]]

a[n_]:=Sum[If[p[n, k], 1, 0], {k, 1, (n-1)/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

Zhi-Wei Sun, Feb 02 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 6 16:34 EDT 2020. Contains 334828 sequences. (Running on oeis4.)