A237523 a(n) = |{0 < k < n/2: phi(k*(n-k)) + 1 is a square}|, where phi(.) is Euler's totient function.

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 2, 1, 2, 1, 2, 5, 4, 4, 1, 3, 3, 3, 2, 4, 4, 4, 2, 4, 5, 6, 5, 3, 3, 3, 6, 5, 4, 4, 6, 6, 2, 6, 6, 6, 2, 6, 5, 5, 2, 4, 4, 7, 7, 4, 3, 5, 5, 9, 5, 5, 3, 5, 2, 3, 10, 10, 9, 7, 5, 8, 5, 9, 8, 6, 4, 5, 6, 11, 5

Offset

1,10

Comments

Conjecture: (i) a(n) > 0 for all n > 7, and a(n) = 1 only for n = 8, 9, 13, 15, 20, 132.

(ii) If n > 5 is not among 10, 15, 20, 60, 105, then phi(k*(n-k)) is a square for some 0 < k < n/2.

See also A237524 for a similar conjecture involving higher powers.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(8) = 1 since phi(3*5) + 1 = 8 + 1 = 3^2.
a(9) = 1 since phi(4*5) + 1 = 8 + 1 = 3^2.
a(13) = 1 since phi(3*10) + 1 = 8 + 1 = 3^2.
a(15) = 1 since phi(7*8) + 1 = 24 + 1 = 5^2.
a(20) = 1 since phi(6*14) + 1 = 24 + 1 = 5^2.
a(132) = 1 since phi(46*(132-46)) + 1 = 1848 + 1 = 43^2.

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
s[n_]:=SQ[EulerPhi[n]+1]
a[n_]:=Sum[If[s[k(n-k)], 1, 0], {k, 1, (n-1)/2}]
Table[a[n], {n, 1, 80}]

Crossrefs

Cf. A000010, A000290, A234246, A236998, A237497, A237524.

Sequence in context: A195150 A022307 A029413 * A339812 A238568 A368148

Adjacent sequences: A237520 A237521 A237522 * A237524 A237525 A237526

Keyword

nonn

Author

Zhi-Wei Sun, Feb 08 2014

Status

approved