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A236567 a(n) = |{0 < k < n - 2: k + phi(n-k)/2 is a square}|, where phi(.) is Euler's totient function. 3
0, 0, 0, 0, 0, 1, 2, 1, 1, 2, 1, 3, 1, 2, 4, 1, 1, 2, 4, 3, 2, 3, 1, 3, 2, 3, 3, 4, 3, 1, 8, 3, 3, 2, 4, 4, 2, 2, 5, 6, 4, 6, 3, 2, 5, 4, 4, 5, 4, 1, 8, 6, 3, 3, 5, 6, 3, 4, 5, 9, 5, 2, 3, 6, 6, 5, 4, 4, 6, 8, 6, 8, 4, 3, 5, 8, 4, 1, 6, 6, 6, 3, 9, 8, 5, 4, 6, 7, 7, 6, 7, 5, 6, 8, 3, 10, 5, 5, 5, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

Conjecture: (i) a(n) > 0 for all n > 5.

(ii) For n > 31, there is a positive integer k < n - 2 with phi(k) + phi(n-k)/2 a square. If n > 70 is not equal to 150, then phi(k) + phi(n-k) is a square for some 0 < k < n.

(iii) If n > 5, then phi(k) + phi(n-k)/2 is a triangular number for some 0 < k < n - 2. For each n = 20, 21, ... there is a positive integer k < n with (phi(k) + phi(n-k))/2 a triangular number.

LINKS

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

EXAMPLE

a(8) = 1 since 1 + phi(7)/2 = 1 + 3 = 2^2.

a(11) = 1 since 8 + phi(3)/2 = 8 + 1 = 3^2.

a(78) = 1 since 40 + phi(38)/2 = 40 + 9 = 7^2.

MATHEMATICA

SQ[n_]:=IntegerQ[Sqrt[n]]

p[n_, k_]:=SQ[k+EulerPhi[n-k]/2]

a[n_]:=Sum[If[p[n, k], 1, 0], {k, 1, n-3}]

Table[a[n], {n, 1, 100}]

CROSSREFS

Cf. A000010, A000290, A234246.

Sequence in context: A050305 A117164 A212182 * A247299 A127586 A055893

Adjacent sequences:  A236564 A236565 A236566 * A236568 A236569 A236570

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 02 2014

STATUS

approved

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Last modified September 29 03:38 EDT 2020. Contains 337420 sequences. (Running on oeis4.)