A134541 -id:A134541 - cp's OEIS frontend

A002321 Mertens's function: Sum_{k=1..n} mu(k), where mu is the Moebius function A008683.

1, 0, -1, -1, -2, -1, -2, -2, -2, -1, -2, -2, -3, -2, -1, -1, -2, -2, -3, -3, -2, -1, -2, -2, -2, -1, -1, -1, -2, -3, -4, -4, -3, -2, -1, -1, -2, -1, 0, 0, -1, -2, -3, -3, -3, -2, -3, -3, -3, -3, -2, -2, -3, -3, -2, -2, -1, 0, -1, -1, -2, -1, -1, -1, 0, -1, -2, -2, -1, -2, -3, -3, -4, -3, -3, -3, -2, -3, -4, -4, -4

Offset: 1

N. J. A. Sloane

Partial sums of the Moebius function A008683.
Also determinant of n X n (0,1) matrix defined by A(i,j)=1 if j=1 or i divides j.
The first positive value of Mertens's function for n > 1 is for n = 94. The graph seems to show a negative bias for the Mertens function which is eerily similar to the Chebyshev bias (described in A156749 and A156709). The purported bias seems to be empirically approximated to - (6 / Pi^2) * (sqrt(n) / 4) (by looking at the graph) (see MathOverflow link, May 28 2012) where 6 / Pi^2 = 1 / zeta(2) is the asymptotic density of squarefree numbers (the squareful numbers having Moebius mu of 0). This would be a growth pattern akin to the Chebyshev bias. - Daniel Forgues, Jan 23 2011
All integers appear infinitely often in this sequence. - Charles R Greathouse IV, Aug 06 2012
Soundararajan proves that, on the Riemann Hypothesis, a(n) << sqrt(n) exp(sqrt(log n)*(log log n)^14), sharpening the well-known equivalence. - Charles R Greathouse IV, Jul 17 2015
Balazard & De Roton improve this (on the Riemann Hypothesis) to a(n) << sqrt(n) exp(sqrt(log n)*(log log n)^k) for any k > 5/2, where the implied constant in the Vinogradov symbol depends on k. Saha & Sankaranarayanan reduce the exponent to 5/4 on additional hypotheses. - Charles R Greathouse IV, Feb 02 2023

			G.f. = x - x^3 - x^4 - 2*x^5 - x^6 - 2*x^7 - 2*x^8 - 2*x^9 - x^10 - 2*x^11 - 2*x^12 - ...

E. Landau, Vorlesungen über Zahlentheorie, Chelsea, NY, Vol. 2, p. 157.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
F. Mertens, "Über eine zahlentheoretische Funktion", Akademie Wissenschaftlicher Wien Mathematik-Naturlich Kleine Sitzungsber, IIa 106, (1897), p. 761-830.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VI.1.
Biswajyoti Saha and Ayyadurai Sankaranarayanan, On estimates of the Mertens function, International Journal of Number Theory, Vol. 15, No. 02 (2019), pp. 327-337.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. von zur Gathen and J. Gerhard, Modern Computer Algebra, Cambridge, 1999, see p. 482.

T. D. Noe, Table of n, a(n) for n = 1..10000
Michel Balazard and Anne De Roton, Sur un critère de Baez-Duarte pour l'hypothèse de Riemann, International Journal of Number TheoryVol. 06, No. 04, pp. 883-903 (2010). arXiv preprint (2008). arXiv:0812.1689 [math.NT]
B. Boncompagni, Selected values of the Mertens function.
O. Bordelles, Some Explicit Estimates for the Mobius Function , J. Int. Seq. 18 (2015), 15.11.1.
G. J. Chaitin, Thoughts on the Riemann hypothesis, arXiv:math/0306042 [math.HO], 2003.
J. B. Conrey, The Riemann Hypothesis, Notices Amer. Math. Soc., 50 (No. 3, March 2003), 341-353. See p. 347.
Marc Deléglise and Joël Rivat, Computing the summation of the Mobius function, Experiment. Math. 5:4 (1996), pp. 291-295.
F. Dress, Fonction sommatoire de la fonction de Moebius. 1. Majorations expérimentales, Experiment. Math. , Volume 2, Issue 2 (1993), 89-98.
F. Dress and M. El Marraki, Fonction sommatoire de la fonction de Moebius. 2. Majorations asymptotiques élémentaires, Experiment. Math., Volume 2, Issue 2 (1993), 99-112.
M. El-Marraki, Fonction sommatoire de la fonction mu de Möbius, 3. Majorations asymptotiques effectives fortes, Journal de théorie des nombres de Bordeaux, Tome 7 (1995) no. 2 , p. 407-433.
Brady Haran, Holly Krieger, and Pete McPartlan, A Prime Surprise (Mertens Conjecture), Numberphile video (2019).
Harald A. Helfgott and Lola Thompson, Summing mu(n): a faster elementary algorithm, arXiv:2101.08773 [math.NT], 2021.
Greg Hurst, Computations of the Mertens function and improved bounds on the Mertens conjecture, arXiv:1610.08551 [math.NT], 2016-2017.
MathOverflow, Is Mertens function negatively biased?, posted May 28, 2012.
MathOverflow, Approximations to the Mertens function, posted Jul 08 2015.
Nathan Ng, The distribution of the summatory function of the Möbius function, Proc. London Math. Soc. (3) 89 (2004), no. 2, 361-389; arXiv:math/0310381 [math.NT], 2003.
A. M. Odlyzko and H. J. J. te Riele, Disproof of the Mertens conjecture, J. reine angew. Math., 357 (1985), pp. 138-160.
Lowell Schoenfeld, An improved estimate for the summatory function of the Möbius function, Acta Arithmetica 15:3 (1969), pp. 221-233.
Kannan Soundararajan, Partial sums of the Möbius function, Journal für die reine und angewandte Mathematik, Vol. 631 (2009), pp. 141-152. arXiv:0705.0723 [math.NT], 2007-2008.
Robert Daublebsky von Sterneck, Empirische Untersuchung ueber den Verlauf der zahlentheoretischer Function sigma(n) = Sum_{x=1..n} mu(x) im Intervalle von 0 bis 150 000, Sitzungsbericht der Kaiserlichen Akademie der Wissenschaften Wien, Mathematisch-Naturwissenschaftlichen Klasse, 2a, v. 106, 1897, 835-1024.
Robert Daublebsky von Sterneck, Bemerkung über die Summierung einiger zahlen-theoretischen Functionen, Monatshefte für Mathematik und Physik 9(1) (1898), 43-45. [He proves the inequality |a(n)| <= (n/9) + 8.]
Paul Tarau, Towards a generic view of primality through multiset decompositions of natural numbers, Theoretical Computer Science, Volume 537, Jun 05 2014, Pages 105-124.
Paul Tarau, Emulating Primality with Multiset Representations of Natural Numbers, in Theoretical Aspects Of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238.
G. Villemin's Almanac of Numbers, Nombres de Moebius et de Mertens.
Eric Weisstein's World of Mathematics, Mertens Function
Eric Weisstein's World of Mathematics, Redheffer Matrix.
Wikipedia, Mertens function.
H. S. Wilf, A Greeting; and a view of Riemann's Hypothesis, Amer. Math. Monthly, 94:1 (1987), 3-6.

Cf. A008683, A059571, A084237, A209802.
First column of A134541.
First column of A179287.

import Data.List (genericIndex)
a002321 n = genericIndex a002321_list (n-1)
a002321_list = scanl1 (+) a008683_list
-- Reinhard Zumkeller, Jul 14 2014, Dec 26 2012

[&+[MoebiusMu(k): k in [1..n]]: n in [1..81]]; // Bruno Berselli, Jul 12 2021

with(numtheory); A002321 := n->add(mobius(k),k=1..n);

Rest[ FoldList[ #1+#2&, 0, Array[ MoebiusMu, 100 ] ] ]
Accumulate[Array[MoebiusMu,100]] (* Harvey P. Dale, May 11 2011 *)

PARI
```
a(n) = sum( k=1, n, moebius(k))
```

a(n) = if( n<1, 0, matdet( matrix(n, n, i, j, j==1 || 0==j%i)))

a(n)=my(s); forsquarefree(k=1,n, s+=moebius(k)); s \\ Charles R Greathouse IV, Jan 08 2018

from sympy import mobius
def M(n): return sum(mobius(k) for k in range(1,n + 1))
print([M(n) for n in range(1, 151)]) # Indranil Ghosh, Mar 18 2017

from functools import lru_cache
@lru_cache(maxsize=None)
def A002321(n):
    if n == 0:
        return 0
    c, j = n, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A002321(k1)
        j, k1 = j2, n//j2
    return j-c # Chai Wah Wu, Mar 30 2021

Assuming the Riemann hypothesis, a(n) = O(x^(1/2 + eps)) for every eps > 0 (Littlewood - see Landau p. 161).
Lambert series: Sum_{n >= 1} a(n)*(x^n/(1-x^n)-x^(n+1)/(1-x^(n+1))) = x and -1/x. - Mats Granvik, Sep 09 2010 and Sep 23 2010
a(n)+2 = A192763(n,1) for n>1, and A192763(1,k) for k>1 (conjecture). - Mats Granvik, Jul 10 2011
Sum_{k = 1..n} a(floor(n/k)) = 1. - David W. Wilson, Feb 27 2012
a(n) = Sum_{k = 1..n} tau_{-2}(k) * floor(n/k), where tau_{-2} is A007427. - Enrique Pérez Herrero, Jan 23 2013
a(n) = Sum_{k=1..A002088(n)} exp(2*Pi*i*A038566(k)/A038567(k-1)) where i is the imaginary unit. - Eric Desbiaux, Jul 31 2014
Schoenfeld proves that |a(n)| < 5.3*n/(log n)^(10/9) for n > 1. - Charles R Greathouse IV, Jan 17 2018
G.f. A(x) satisfies: A(x) = (1/(1 - x)) * (x - Sum_{k>=2} (1 - x^k) * A(x^k)). - Ilya Gutkovskiy, Aug 11 2021

A143327 Table T(n,k) by antidiagonals. T(n,k) is the number of primitive (=aperiodic) k-ary words (n,k >= 1) with length less than or equal to n which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 11, 11, 1, 1, 5, 19, 35, 26, 1, 1, 6, 29, 79, 115, 53, 1, 1, 7, 41, 149, 334, 347, 116, 1, 1, 8, 55, 251, 773, 1339, 1075, 236, 1, 1, 9, 71, 391, 1546, 3869, 5434, 3235, 488, 1, 1, 10, 89, 575, 2791, 9281, 19493, 21754, 9787, 983, 1, 1, 11

Offset: 1

Alois P. Heinz, Aug 07 2008

The coefficients of the polynomial of row n are given by the n-th row of triangle A134541; for example row 4 has polynomial -1+k^2+k^3.

			T(3,3) = 11, because 11 words of length <=3 over 3-letter alphabet {a,b,c} are primitive and earlier than others derived by cyclic shifts of the alphabet: a, ab, ac, aab, aac, aba, abb, abc, aca, acb, acc.
Table begins:
  1,   1,    1,     1,     1,      1,      1,       1, ...
  1,   2,    3,     4,     5,      6,      7,       8, ...
  1,   5,   11,    19,    29,     41,     55,      71, ...
  1,  11,   35,    79,   149,    251,    391,     575, ...
  1,  26,  115,   334,   773,   1546,   2791,    4670, ...
  1,  53,  347,  1339,  3869,   9281,  19543,   37367, ...
  1, 116, 1075,  5434, 19493,  55936, 137191,  299510, ...
  1, 236, 3235, 21754, 97493, 335656, 960391, 2396150, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Index entries for sequences related to Lyndon words

Columns k=1-10 give: A000012, A085945, A320087, A320088, A320089, A320090, A320091, A320092, A320093, A320094.
Rows n=1-4 give: A000012, A000027, A028387, A003777.
Main diagonal gives A320095.
Cf. A143325, A143326, A134541, A008683.

with(numtheory):
f1:= proc (n) option remember; unapply(k^(n-1)
        -add(f1(d)(k), d=divisors(n) minus {n}), k)
     end:
g1:= proc(n) option remember; unapply(add(f1(j)(x), j=1..n), x) end:
T:= (n, k)-> g1(n)(k):
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);

t[n_, k_] := Sum[k^(d-1)*MoebiusMu[j/d], {j, 1, n}, {d, Divisors[j]}]; Table[t[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 13 2013 *)

T(n,k) = Sum_{j=1..n} Sum_{d|j} k^(d-1) * mu(j/d).
T(n,k) = Sum_{j=1..n} A143325(j,k).
T(n,k) = A143326(n,k) / k.

A143326 Table T(n,k) by antidiagonals. T(n,k) is the number of primitive (=aperiodic) k-ary words with length less than or equal to n (n,k >= 1).

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 9, 10, 1, 5, 16, 33, 22, 1, 6, 25, 76, 105, 52, 1, 7, 36, 145, 316, 345, 106, 1, 8, 49, 246, 745, 1336, 1041, 232, 1, 9, 64, 385, 1506, 3865, 5356, 3225, 472, 1, 10, 81, 568, 2737, 9276, 19345, 21736, 9705, 976, 1, 11, 100, 801, 4600, 19537, 55686

Offset: 1

Alois P. Heinz, Aug 07 2008

The coefficients of the polynomial of row n are given by the n-th row of triangle A134541; for example row 4 has polynomial -k+k^3+k^4.

			T(2,3) = 9, because there are 9 primitive words of length less than or equal to 2 over 3-letter alphabet {a,b,c}: a, b, c, ab, ac, ba, bc, ca, cb; note that the non-primitive words aa, bb and cc don't belong to the list; secondly note that the words in the list need not be Lyndon words, for example ba can be derived from ab by a cyclic rotation of the positions.
Table begins:
  1,   2,    3,     4,      5,       6,       7,        8, ...
  1,   4,    9,    16,     25,      36,      49,       64, ...
  1,  10,   33,    76,    145,     246,     385,      568, ...
  1,  22,  105,   316,    745,    1506,    2737,     4600, ...
  1,  52,  345,  1336,   3865,    9276,   19537,    37360, ...
  1, 106, 1041,  5356,  19345,   55686,  136801,   298936, ...
  1, 232, 3225, 21736,  97465,  335616,  960337,  2396080, ...
  1, 472, 9705, 87016, 487465, 2013936, 6722737, 19169200, ...
  ...
From _Wolfdieter Lang_, Feb 01 2014: (Start)
The triangle Tri(n,m) := T(m,n-(m-1)) begins:
n\m  1   2    3     4     5      6      7     8    9  10 ...
1:   1
2:   2   1
3:   3   4    1
4:   4   9   10     1
5:   5  16   33    22     1
6:   6  25   76   105    52      1
7:   7  36  145   316   345    106      1
8:   8  49  246   745  1336   1041    232     1
9:   9  64  385  1506  3865   5356   3225   472    1
10: 10  81  568  2737  9276  19345  21736  9705  976   1
...
For the columns see A000027, A000290, A081437, ... (End)

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Index entries for sequences related to Lyndon words

Column 1: A000012. Rows 1-3: A000027, A000290, A081437 and A085490. See also A143324, A143327, A134541, A008683.

with(numtheory):
f0:= proc(n) option remember;
       unapply(k^n-add(f0(d)(k), d=divisors(n) minus {n}), k)
     end:
g0:= proc(n) option remember; unapply(add(f0(j)(x), j=1..n), x) end:
T:= (n, k)-> g0(n)(k):
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);

f0[n_] := f0[n] = Function[k, k^n-Sum[f0[d][k], {d, Divisors[n] // Most}]]; g0[n_] := g0[n] = Function[x, Sum[f0[j][x], {j, 1, n}]]; T[n_, k_] := g0[n][k]; Table[T[n, 1+d-n], {d, 1, 12}, {n, 1, d}]//Flatten (* Jean-François Alcover, Feb 12 2014, translated from Maple *)

T(n,k) = Sum_{1<=j<=n} Sum_{d|j} k^d * mu(j/d).
T(n,k) = Sum_{1<=j<=n} A143324(j,k).
T(n,k) = A143327(n,k) * k.

cp's OEIS Frontend

A134542 A134541 * A000012.

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A002321 Mertens's function: Sum_{k=1..n} mu(k), where mu is the Moebius function A008683.

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A143327 Table T(n,k) by antidiagonals. T(n,k) is the number of primitive (=aperiodic) k-ary words (n,k >= 1) with length less than or equal to n which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

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A143326 Table T(n,k) by antidiagonals. T(n,k) is the number of primitive (=aperiodic) k-ary words with length less than or equal to n (n,k >= 1).

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A176702 Triangle T(n,k) read by rows. A051731(n,k)-A051731(n,k+1).

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