The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A237016 a(n) = |{0 < k < n: phi(k)*sigma(n-k) is a square}|, where phi(.) is Euler's totient function and sigma(j) is the sum of all positive divisors of j. 4
 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 3, 1, 4, 2, 2, 1, 0, 1, 2, 2, 2, 2, 6, 4, 2, 2, 4, 2, 2, 4, 1, 6, 5, 6, 3, 3, 8, 3, 2, 4, 6, 1, 2, 4, 3, 3, 3, 5, 6, 5, 5, 3, 2, 5, 4, 4, 3, 6, 5, 7, 10, 7, 4, 2, 1, 4, 6, 7, 9, 6, 12, 3, 3, 4, 12, 6, 6, 5, 6, 4, 5, 8, 6, 5, 10, 7, 7, 2, 5, 8, 4, 2, 4, 3, 8, 4, 4, 11, 6, 6 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,11 COMMENTS Conjecture: (i) a(n) > 0 except for n = 1, 7, 17. (ii) If n > 5, then phi(k)*sigma(n-k) + 1 is a square for some 0 < k < n. (iii) If n > 309, then there is a positive integer k < n/2 such that sigma(k)*sigma(n-k) is a square. See also A236998 for a similar conjecture. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 EXAMPLE a(9) = 1 since phi(8)*sigma(1) = 4*1 = 2^2. a(16) = 1 since phi(6)*sigma(10) = 2*18 = 6^2. a(31) = 1 since phi(24)*sigma(7) = 8*8 = 8^2. a(65) = 1 since phi(19)*sigma(46) = 18*72 = 36^2. MATHEMATICA sigma[n_]:=DivisorSigma[1, n] SQ[n_]:=IntegerQ[Sqrt[n]] p[n_, k_]:=SQ[EulerPhi[k]*sigma[n-k]] a[n_]:=Sum[If[p[n, k], 1, 0], {k, 1, n-1}] Table[a[n], {n, 1, 100}] CROSSREFS Cf. A000010, A000203, A000290, A234246, A236567, A236998. Sequence in context: A283866 A134545 A174907 * A135822 A242111 A209919 Adjacent sequences: A237013 A237014 A237015 * A237017 A237018 A237019 KEYWORD nonn AUTHOR Zhi-Wei Sun, Feb 02 2014 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 13 03:33 EDT 2024. Contains 375857 sequences. (Running on oeis4.)