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A237523
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a(n) = |{0 < k < n/2: phi(k*(n-k)) + 1 is a square}|, where phi(.) is Euler's totient function.
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3
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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
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OFFSET
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1,10
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COMMENTS
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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.
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LINKS
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Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
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EXAMPLE
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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.
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MATHEMATICA
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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}]
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CROSSREFS
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Cf. A000010, A000290, A234246, A236998, A237497, A237524.
Sequence in context: A195150 A022307 A029413 * A339812 A238568 A238421
Adjacent sequences: A237520 A237521 A237522 * A237524 A237525 A237526
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KEYWORD
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nonn
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AUTHOR
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Zhi-Wei Sun, Feb 08 2014
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STATUS
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approved
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