1,1

In contrast to sister sequences A083250 and A083252, the terms here must be even. Writing m = 2^k * s with k > 0 and odd s, we get a bound s - phi(s) <= 2*tau(s). On the other hands, for a composite odd s, we have s - phi(s) >= s/3, while s/3 <= 2*tau(s) < 4*sqrt(s) implies that s < 144. By direct check, we get that m = 72 (with s=9 and k=3), is the only term with composite s. For noncomposite s, we get a standalone term m = 2, and a series m = 16 * p for any prime p > 2. - Max Alekseyev, Nov 17 2025

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

For n >= 4, a(n) = 16 * prime(n-1), as originally proved by Lawrence Sze. - Ralf Stephan, Nov 16 2004; updated by Max Alekseyev, Nov 17 2025

For n=2896: d=10 divisors, r=1440 coprimes, u=1447 unrelated or n - u = r + d - 1 = 1449 related numbers to n; thus abs(1449 - 1447) = 2.

Do[r=EulerPhi[n]; d=DivisorSigma[0, n]; u=n-r-d+1; df=2*u-n; If[Equal[Abs[df], 2], Print[n(*, {d, r, u}*)]], {n, 1, 3000}]

Cf. A000005, A000010, A045763, A073757, A083243-A083249, A083250, A083252.

Sequence in context: A087259 A195876 A326778 * A226401 A226396 A341110

Adjacent sequences: A083248 A083249 A083250 * A083252 A083253 A083254

easy,nonn

Labos Elemer, May 07 2003

approved