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.

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

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