A038199 -id:A038199 - cp's OEIS frontend

A119920 Number of rationals in [0, 1) having exactly n preperiodic bits, then exactly n periodic bits.

1, 4, 24, 96, 480, 1728, 8064, 30720, 129024, 506880, 2095104, 8232960, 33546240, 133152768, 536248320, 2139095040, 8589803520, 34285289472, 137438429184, 549212651520, 2198882746368, 8791793860608, 35184363700224

Offset: 1

Brad Chalfan (brad(AT)chalfan.net), May 28 2006

			The binary expansion of 7/24 = 0.010(01)... has 3 preperiodic bits (to the right of the binary point) followed by 2 periodic (i.e., repeating) bits, while 1/2 = 0.1(0)... has one bit of each type. The preperiodic and periodic parts are both chosen to be as short as possible.
a(2) = |{1/12 = 0.00(01)..., 5/12 = 0.01(10)..., 7/12 = 0.10(01)..., 11/12 = 0.11(10)...}| = 4

Elementwise product of 2^n (offset 0) and A038199. Also, diagonal of A119918.

Table[2^(n-1)(Plus@@((2^Divisors[n]-1)MoebiusMu[n/Divisors[n]])),{n,1,23} ]

a(n) = 2^(n-1) * sum_{d|n} (2^d - 1) * mu(n/d)

A346558 a(n) = Sum_{d|n} phi(n/d) * (2^d - 1).

Original entry on oeis.org

1, 4, 9, 20, 35, 78, 133, 280, 531, 1070, 2057, 4212, 8203, 16534, 32865, 65840, 131087, 262818, 524305, 1049740, 2097459, 4196390, 8388629, 16782024, 33554575, 67117102, 134218809, 268452212, 536870939, 1073777010, 2147483677, 4295033440, 8589938775, 17180000318, 34359739085

Offset: 1

Ilya Gutkovskiy, Sep 17 2021

nonn

Cf. A000010, A000031, A000225, A008965, A034738, A038199, A053635.

Table[Sum[EulerPhi[n/d] (2^d - 1), {d, Divisors[n]}], {n, 1, 35}]
nmax = 35; CoefficientList[Series[Sum[EulerPhi[k] x^k/((1 - x^k) (1 - 2 x^k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, eulerphi(n/d)*(2^d - 1)); \\ Michel Marcus, Sep 17 2021

G.f.: Sum_{k>=1} phi(k) * x^k / ((1 - x^k) * (1 - 2*x^k)).
a(n) = Sum_{k=1..n} (2^gcd(n,k) - 1).
a(n) = n * (A000031(n) - 1) = n * A008965(n).
Dirichlet convolution of A000225 and A000010. - R. J. Mathar, Sep 30 2021

cp's OEIS Frontend

A119920 Number of rationals in [0, 1) having exactly n preperiodic bits, then exactly n periodic bits.

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