A038199 Row sums of triangle T(m,n) = number of solutions to 1 <= a(1) < a(2) < ... < a(m) <= n, where gcd(a(1), a(2), ..., a(m), n) = 1, in A020921.

1, 2, 6, 12, 30, 54, 126, 240, 504, 990, 2046, 4020, 8190, 16254, 32730, 65280, 131070, 261576, 524286, 1047540, 2097018, 4192254, 8388606, 16772880, 33554400, 67100670, 134217216, 268419060, 536870910, 1073708010, 2147483646

Offset

1,2

Comments

The function T(m,n) described above has an inverse: see A038200.

Also, Moebius transform of 2^n - 1 = A000225. Also, number of rationals in [0, 1) whose binary expansions consist just of repeating bits of (least) period exactly n (i.e., there's no preperiodic part), where 0 = 0.000... is considered to have period 1. - Brad Chalfan (brad(AT)chalfan.net), May 29 2006

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Henk Bruin, C. Carminati, and C. Kalle, Matching for generalised beta-transformations, arXiv preprint arXiv:1610.01872 [math.DS], 2016.

Henk Bruin, C. Carminati, and C. Kalle, Matching for generalised beta-transformations, Indagationes Mathematicae 28 (2017), 55-73.

M. B. Nathanson, Primitive sets and Euler phi function for subsets of {1,2,...,n}, arXiv:math/0608150 [math.NT], 2006-2007.

Prapanpong Pongsriiam, Relatively Prime Sets, Divisor Sums, and Partial Sums, arXiv:1306.4891 [math.NT], 2013 and J. Int. Seq. 16 (2013) #13.9.1.

P. Pongsriiam, A remark on relatively prime sets, Integers 13 (2013), A49.

Temba Shonhiwa, A Generalization of the Euler and Jordan Totient Functions, Fib. Quart., 37 (1999), 67-76.

Wikipedia, Lambert series.

Formula

a(n) = Sum_{d | n} mu(n/d)*(2^d-1). - Paul Barry, Mar 20 2005

Lambert g.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = x/((1 - x)*(1 - 2*x)). - Ilya Gutkovskiy, Apr 25 2017

O.g.f.: Sum_{d >= 1} mu(d)*(x^d/((1 - x^d)*(1 - 2*x^d)). - Petros Hadjicostas, Jun 18 2019

Mathematica

Table[Plus@@((2^Divisors[n]-1)MoebiusMu[n/Divisors[n]]), {n, 1, 31}] (* Brad Chalfan (brad(AT)chalfan.net), May 29 2006 *)

Prog

(Haskell)
a038199 n = sum [a008683 (n `div` d) * (a000225 d)| d <- a027750_row n]
-- Reinhard Zumkeller, Feb 17 2013
(Python)
from sympy import mobius, divisors
def a(n): return sum(mobius(n//d) * (2**d - 1) for d in divisors(n))
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jun 28 2017
(PARI) a(n) = sumdiv(n, d, moebius(n/d)*(2^d-1)); \\ Michel Marcus, Jun 28 2017

Crossrefs

A027375, A038199 and A056267 are all essentially the same sequence with different initial terms.

Cf. A000225, A008683, A020921, A023995, A027750, A038200, A130887.

Cf. A059966 (a(n)/n).

Sequence in context: A024701 A224532 A179674 * A056267 A348412 A133996

Adjacent sequences: A038196 A038197 A038198 * A038200 A038201 A038202

Keyword

nonn,easy,nice

Author

Temba Shonhiwa (Temba(AT)maths.uz.ac.zw)

Extensions

Better description from Michael Somos

More terms from Naohiro Nomoto, Sep 10 2001

More terms from Brad Chalfan (brad(AT)chalfan.net), May 29 2006

Status

approved