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

Author

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

Keywords

• nonn
• easy
• nice

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, Carlo Carminati, and Charlene Kalle, Matching for generalised beta-transformations, arXiv preprint arXiv:1610.01872 [math.DS], 2016.
• Henk Bruin, Carlo Carminati, and Charlene Kalle, Matching for generalised beta-transformations, Indagationes Mathematicae 28 (2017), 55-73.
• Jason D. Chadwick, Mariesa H. Teo, Joshua Viszlai, Willers Yang, and Frederic T. Chong, Erasure Minesweeper: exploring hybrid-erasure surface code architectures for efficient quantum error correction, arXiv:2505.00066 [quant-ph], 2025. See p. 14.
• Melvyn 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.
• Prapanpong 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.

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).

Programs

Haskell

a038199 n = sum [a008683 (n `div` d) * (a000225 d)| d <- a027750_row n]
-- Reinhard Zumkeller, Feb 17 2013

Mathematica

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

PARI

a(n) = sumdiv(n, d, moebius(n/d)*(2^d-1)); \\ Michel Marcus, Jun 28 2017

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

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

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