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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
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
Sequence in context: A050305 A117164 A212182 * A360564 A247299 A127586
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 02 2014
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)