A130887 -id:A130887 - cp's OEIS frontend

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

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

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

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.

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).

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

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

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

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

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

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

A198383 a(n) = Sum_{k=1..n} 2^(n mod k).

Original entry on oeis.org

1, 2, 4, 5, 10, 10, 20, 22, 37, 40, 80, 72, 144, 158, 278, 283, 566, 548, 1096, 1120, 2106, 2162, 4324, 4210, 8389, 8584, 16650, 16772, 33544, 33194, 66388, 66968, 131882, 132690, 265222, 263607, 527214, 530138, 1052078, 1054254, 2108508, 2103282, 4206564, 4216760

Offset: 1

Benoit Cloitre, Oct 24 2011

nonn

A more precise asymptotic formula is given in the link.
From David Morales Marciel, Oct 19 2015: (Start)
If n is prime then a(n)=2*a(n-1).
It appears that for every (deficient, abundant)-pair of numbers (11+6x, 11+6x+1), a(11+6x) > a(11+6x+1).
(End)

Amiram Eldar, Table of n, a(n) for n = 1..1000
Benoit Cloitre, An asymptotic formula for sum_{k=1..n}x^(n mod k) [broken link, draft]

Cf. A198259, A000079, A130887.

Table[Sum[2^Mod[n, k], {k, n}], {n, 44}] (* Michael De Vlieger, Oct 19 2015 *)

PARI
```
a(n) = sum(k=1, n, 2^(n%k))
```

a(n) = 2^ceiling(n/2) + O(2^(n/3)).

a(n) = 2*a(n-1) + A000079(n) - A130887(n). - Ridouane Oudra, May 03 2025

A130888 Triangle read by rows, A051731(n,k) dot (1, 3, 7, 15, ...) with like numbers of terms.

Original entry on oeis.org

1, 1, 3, 1, 0, 7, 1, 3, 0, 15, 1, 0, 0, 0, 31, 1, 3, 7, 0, 0, 63, 1, 0, 0, 0, 0, 0, 127, 1, 3, 0, 15, 0, 0, 0, 255

Offset: 1

Gary W. Adamson, Jun 07 2007

Row sums = A130887: (1, 4, 8, 19, 32, 74, 128, 274, 519, 1058, ...).

			First few rows of the triangle:
  1;
  1,  3;
  1,  0,  7;
  1,  3,  0, 15;
  1,  0,  0,  0, 31;
  1,  3,  7,  0,  0, 63;
  ...
Row 4 = (1, 3, 0, 15) = (1, 1, 0, 1) dot (1, 3, 7, 15); where (1, 1, 0, 1) = row 4 of A051731.

Cf. A051731, A130887.

Triangle read by rows, dot product of each row of A051731 and the same number of terms in the series (1, 3, 7, 15, ...).

G.f.: Sum_{k>0} (2^k-1)*(x*y)^k/(1-x^k). - Vladeta Jovovic, Dec 02 2008

cp's OEIS Frontend

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.

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A198383 a(n) = Sum_{k=1..n} 2^(n mod k).

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Programs

Mathematica

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A130888 Triangle read by rows, A051731(n,k) dot (1, 3, 7, 15, ...) with like numbers of terms.

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Author

Keywords

Comments

Examples

Crossrefs

Formula