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 A234246 a(n) = |{0 < k < n: k*phi(n-k) + 1 is a square}|, where phi(.) is Euler's totient function. 10
 0, 0, 0, 1, 1, 0, 2, 1, 1, 3, 2, 1, 1, 2, 3, 4, 5, 4, 2, 2, 2, 5, 4, 1, 5, 4, 4, 3, 2, 8, 5, 2, 1, 3, 9, 5, 9, 4, 4, 6, 2, 4, 9, 5, 5, 7, 9, 3, 1, 10, 6, 8, 3, 6, 4, 5, 7, 8, 3, 5, 5, 4, 6, 6, 10, 14, 8, 3, 3, 6, 9, 5, 7, 7, 9, 2, 8, 8, 9, 5, 6, 6, 6, 8, 9, 7, 9, 4, 5, 9, 10, 8, 8, 7, 14, 9, 5, 7, 6, 10 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS Conjecture: (i) a(n) > 0 if n is not a divisor of 6. The only values of n with a(n) = 1 are 4, 5, 8, 9, 12, 13, 24, 33, 49. (ii) If n >= 60, then k + phi(n-k) is a square for some 0 < k < n. If n > 60, then sigma(k) + phi(n-k) is a square for some 0 < k < n, where sigma(k) is the sum of all positive divisors of k. (iii) If n > 7 is not equal to 10 or 20, then phi(k)*phi(n-k) + 1 is a square for some 0 < k < n. (iv) If n > 7 is not equal to 10 or 19, then (phi(k) + phi(n-k))/2 is a triangular number for some 0 < k < n. Note that (n - 1)*phi(1) + 1 = n. So a(n) > 0 if n is a square. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 EXAMPLE a(4) = 1 since 3*phi(1) + 1 = 2^2. a(5) = 1 since 3*phi(2) + 1 = 2^2. a(8) = 1 since 4*phi(4) + 1 = 3^2. a(9) = 1 since 8*phi(1) + 1 = 3^2. a(12) = 1 since 2*phi(10) + 1 = 3^2. a(13) = 1 since 4*phi(9) + 1 = 5^2. a(14) = 2 since 2*phi(12) + 1 = 3^2 and 6*phi(8) + 1 = 5^2. a(24) = 1 since 12*phi(12) + 1 = 7^2. a(33) = 1 since 3*phi(30) + 1 = 5^2. a(49) = 1 since 48*phi(1) + 1 = 7^2. MATHEMATICA SQ[n_]:=IntegerQ[Sqrt[n]] a[n_]:=Sum[If[SQ[k*EulerPhi[n-k]+1], 1, 0], {k, 1, n-1}] Table[a[n], {n, 1, 100}] CROSSREFS Cf. A000010, A000290, A233542, A233544, A233547, A233566, A233567, A233867, A233918, A234200 Sequence in context: A296774 A066099 A254111 * A006375 A327520 A184441 Adjacent sequences: A234243 A234244 A234245 * A234247 A234248 A234249 KEYWORD nonn AUTHOR Zhi-Wei Sun, Dec 21 2013 STATUS approved

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Last modified February 5 01:38 EST 2023. Contains 360082 sequences. (Running on oeis4.)