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 A236998 a(n) = |{0 < k < n/2: phi(k)*phi(n-k) is a square}|, where phi(.) is Euler's totient function. 8
 0, 0, 1, 0, 0, 1, 2, 0, 2, 2, 1, 1, 2, 1, 1, 1, 1, 3, 3, 2, 2, 4, 3, 1, 3, 1, 3, 1, 1, 2, 2, 1, 4, 4, 3, 3, 1, 1, 5, 2, 3, 7, 2, 5, 3, 4, 3, 2, 7, 3, 2, 3, 4, 6, 2, 1, 7, 5, 3, 2, 2, 4, 4, 2, 6, 4, 3, 5, 5, 7, 4, 3, 2, 6, 4, 2, 7, 5, 5, 4, 4, 2, 4, 8, 2, 7, 5, 7, 3, 3, 8, 6, 7, 5, 7, 3, 9, 3, 7, 5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS Conjecture: (i) a(n) > 0 for all n > 8. (ii) If n > 20, then phi(k)*phi(n-k) + 1 is a square for some 0 < k < n/2. (iii) If n > 1 is not among 4, 7, 60, 199, 267, then k*phi(n-k) is a square for some 0 < k < n. We have verified part (i) of the conjecture for n up to 2*10^6. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 EXAMPLE a(17) = 1 since phi(5)*phi(12) = 4*4 = 4^2. a(24) = 1 since phi(4)*phi(20) = 2*8 = 4^2. a(56) = 1 since phi(8)*phi(48) = 4*16 = 8^2. MATHEMATICA SQ[n_]:=IntegerQ[Sqrt[n]] p[n_, k_]:=SQ[EulerPhi[k]*EulerPhi[n-k]] a[n_]:=Sum[If[p[n, k], 1, 0], {k, 1, (n-1)/2}] Table[a[n], {n, 1, 100}] CROSSREFS Cf. A000010, A000290, A234246, A236567, A237016. Sequence in context: A242667 A059581 A344319 * A297116 A163542 A061895 Adjacent sequences: A236995 A236996 A236997 * A236999 A237000 A237001 KEYWORD nonn AUTHOR Zhi-Wei Sun, Feb 02 2014 STATUS approved

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Last modified September 7 22:13 EDT 2024. Contains 375749 sequences. (Running on oeis4.)