A055464 Numbers n such that sum of EulerPhi and DivisorSum is an integer multiple of the number of divisors.

1, 2, 3, 4, 5, 7, 11, 13, 15, 17, 19, 21, 23, 25, 29, 30, 31, 33, 35, 37, 39, 41, 43, 45, 47, 48, 49, 51, 53, 55, 56, 57, 59, 61, 65, 67, 69, 70, 71, 73, 77, 78, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 102, 103, 105, 107, 109, 110, 111, 113, 115, 119, 121, 123

Offset

1,2

Comments

Makowski proved that phi(n)+Sigma[n] = nd[n] iff n is a prime (see in Sivaramakrishnan, Chapter I, page 8, Theorem 3).

References

Sivaramakrishnan, R. (1989): Classical Theory of Arithmetical Functions Marcel Dekker, Inc., New York-Basel.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Formula

Solutions to Phi[x]+Sigma[x] = kd[x] or A000203(n)+A000010(n) = k*A000005(n), where k is integer.

Example

It is true for all primes and some composites. n = 99, 6 divisors, Sigma = 156, Phi = 60, 156+60 = 216 = 6*36, k = 36.

Mathematica

okQ[n_]:=Divisible[EulerPhi[n]+DivisorSigma[1, n], DivisorSigma[0, n]]
Select[Range[125], okQ]  (* Harvey P. Dale, Mar 06 2011 *)

Prog

(PARI) isok(n) = !((eulerphi(n) + sigma(n)) % numdiv(n)); \\ Michel Marcus, Dec 01 2017

Crossrefs

Cf. A000005, A065387.

Sequence in context: A001087 A191876 A234719 * A139316 A062972 A231878

Adjacent sequences: A055461 A055462 A055463 * A055465 A055466 A055467

Keyword

nonn

Author

Labos Elemer, Jun 27 2000

Status

approved