A028442 -id:A028442 - cp's OEIS frontend

A057639 First differences of zero-sites (A028442) of Mertens's function A002321.

37, 1, 18, 7, 28, 8, 44, 4, 1, 9, 1, 3, 1, 2, 48, 17, 1, 3, 1, 2, 16, 75, 2, 1, 1, 20, 2, 1, 2, 4, 1, 1, 2, 27, 8, 2, 1, 1, 2, 1, 5, 1, 5, 1, 2, 1, 1, 1, 2, 1, 109, 4, 66, 1, 27, 1, 1, 144, 4, 8, 2, 1, 2, 13, 1, 2, 9, 1, 1, 24, 1, 3, 16, 8, 6, 1, 2, 3, 4, 2, 1, 2, 5, 1, 2, 4, 3, 2, 1, 3, 1, 82, 3, 5

Offset: 1

Labos Elemer, Oct 11 2000

nonn

Mertens's function (A002321) is oscillating. The width of its waves is given here.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002321, A013928, A028442, A013940, A051400, A051401, A051402.

Differences[Position[Accumulate[Array[MoebiusMu,1500]],0]//Flatten] (* Harvey P. Dale, Nov 10 2016 *)

lista(kmax) = {my(s = 0, k1 = 2); for(k2 = 3, kmax, s += moebius(k2); if(s == 0, print1(k2 - k1, ", "); k1 = k2));} \\ Amiram Eldar, Jun 09 2024

a(n) = A028442(n+1) - A028442(n).

Offset corrected by Amiram Eldar, Jun 09 2024

A057627 Number of nonsquarefree numbers not exceeding n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 7, 7, 7, 7, 8, 9, 9, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 16, 16, 16, 17, 18, 19, 19, 20, 20, 21, 21, 22, 22, 22, 22, 23, 23, 23, 24, 25, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 28, 29, 29, 29

Offset: 1

Labos Elemer, Oct 10 2000

Number of integers k in A013929 in the range 1 <= k <= n.
This sequence is different from A013940, albeit the first 35 terms are identical.
Asymptotic to k*n where k = 1 - 1/zeta(2) = 1 - 6/Pi^2 = A229099. - Daniel Forgues, Jan 28 2011
This sequence is the sequence of partial sums of A107078 (not of A056170). - Jason Kimberley, Feb 01 2017
Number of partitions of 2n into two parts with the smallest part nonsquarefree. - Wesley Ivan Hurt, Oct 25 2017

			a(36)=13 because 13 nonsquarefree numbers exist which do not exceed 36:{4,8,9,12,16,18,20,24,25,27,28,32,36}.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005117, A008683, A013929, A013940, A013928, A002321, A028442, A107078, A229099, A294242.

N:= 1000: # to get terms up to a(N)
B:= Array(1..N, numtheory:-issqrfree):
C:= map(`if`,B,0,1):
A:= map(round,Statistics:-CumulativeSum(C)):
seq(A[n],n=1..N); # Robert Israel, Jun 03 2014

Accumulate[Table[If[SquareFreeQ[n],0,1],{n,80}]] (* Harvey P. Dale, Jun 04 2014 *)

a(n) = my(s=0); forsquarefree(k=1, sqrtint(n), s += (-1)^(#k[2]~) * (n\k[1]^2)); n - s; \\ Charles R Greathouse IV, May 18 2015; corrected by Daniel Suteu, May 11 2023

from math import isqrt
from sympy import mobius
def A057627(n): return n-sum(mobius(k)*(n//k**2) for k in range(1,isqrt(n)+1)) # Chai Wah Wu, May 10 2024

(define (A057627 n) (- n (A013928 (+ n 1))))

a(n) = n - A013928(n+1) = n - Sum_{k=1..n} mu(k)^2.

G.f.: Sum_{k>=1} (1 - mu(k)^2)*x^k/(1 - x). - Ilya Gutkovskiy, Apr 17 2017

Offset and formula corrected by Antti Karttunen, Jun 03 2014

A062982 Numbers n such that Mertens's function of n (A002321) is divisible by phi(n).

Original entry on oeis.org

1, 2, 39, 40, 58, 65, 93, 101, 145, 149, 150, 159, 160, 163, 164, 166, 214, 231, 232, 235, 236, 238, 254, 329, 331, 332, 333, 353, 355, 356, 358, 362, 363, 364, 366, 393, 401, 403, 404, 405, 407, 408, 413, 414, 419, 420, 422, 423, 424, 425, 427, 428, 537

Offset: 1

Jason Earls, Jul 25 2001

nonn

Except for the initial term, this sequence is the same as A028442, the n for which Mertens's function M(n) is zero. Because phi(n) >= sqrt(n) and M(n) < sqrt(n) for all known n, phi(n) does not divide M(n), except possibility for some extremely large n. Research project: find the least n > 1 with M(n) not zero and phi(n) divides M(n). - T. D. Noe, Jul 28 2005

Harry J. Smith, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Mertens Conjecture

Cf. A002321.

Select[Range[500], Mod[Plus @@ MoebiusMu[Range[#]], EulerPhi[#]] == 0 &] (* Carl Najafi, Aug 17 2011 *)

M(n)=sum(k=1,n,moebius(k)); j=[]; for(n=1,1500, if(Mod(M(n),eulerphi(n))==0,j=concat(j,n))); j

{ n=m=0; for (k=1, 10^9, m+=moebius(k); if (m%eulerphi(k)==0, write("b062982.txt", n++, " ", k); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 15 2009

A118684 Numbers k for which the Mertens function M(k) (A002321) is 1.

Original entry on oeis.org

1, 94, 97, 98, 99, 100, 146, 147, 148, 161, 162, 215, 216, 230, 237, 330, 334, 337, 338, 349, 350, 351, 352, 365, 394, 397, 399, 400, 415, 416, 418, 538, 539, 540, 542, 606, 794, 799, 800, 801, 806, 809, 810, 813, 815, 816, 822, 851, 852, 870, 874, 875

Offset: 1

Yoav Kallus (colonelmustard(AT)gmail.com), May 20 2006

nonn

			M(94) = Sum_{n=1..94} mu(n) = 1.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Mertens Function

Numbers k such that M(k) = b: A171395 (b=-12), A171394 (b=-11), A171393 (b=-10), A171392 (b=-9), A171391 (b=-8), A171383 (b=-7), A171374 (b=-6), A171236 (b=-5), A171235 (b=-4), A171098 (b=-3), A171097 (b=-2), A171096 (b=-1), A028442 (b=0), this sequence (b=1), A171403 (b=2), A171404 (b=3), A171406 (b=4), A171427 (b=5), A171428 (b=6), A171436 (b=7).

Cf. A002321, A051400.

Select[Range[500], Plus @@ MoebiusMu[Range[#]] == 1 &] (* Carl Najafi, Aug 17 2011 *)

isok(n) = sum(k=1, n, moebius(k)) == 1; \\ Michel Marcus, Nov 20 2017

A100766 Numbers for which the values of the Moebius function (A008683) and the Mertens function (A002321) are both 0.

Original entry on oeis.org

40, 150, 160, 164, 232, 236, 332, 333, 356, 363, 364, 404, 405, 408, 414, 420, 423, 424, 425, 428, 608, 636, 637, 796, 812, 824, 825, 850, 884, 896, 904, 916, 920, 1014, 1220, 1256, 1280, 1292, 1300, 1336, 1492, 1519, 1520, 1521, 1524, 1525, 1528, 1532, 1544

Offset: 1

Alonso del Arte, Jan 03 2005

nonn

This sequence is a subset of A100306, numbers for which the values of the Moebius function and the Mertens function agree and, in a different way, a subset of A028442, zeros of the Mertens function. There are no prime numbers in this sequence.

Numbers k such that k-1 and k are consecutive zeros of the Mertens function. - Amiram Eldar, Jun 13 2020

Donovan Johnson, Table of n, a(n) for n = 1..10000
PrimeFan, Esoteric Integer Sequences
PrimeFan, Esoteric Integer Sequences [Cached copy]

Cf. A100306, A100765, A100767.

(* If not already defined *) If[Names["Mertens"] == {}, Mertens[x_] := Plus @@ MoebiusMu[Range[1, x]]]; Select[Range[2500], MoebiusMu[ # ] == 0 && Mertens[ # ] == 0 &]

Offset corrected by Donovan Johnson, Jun 19 2012

A304239 Indices for which the Mertens function A002321 reaches its extremum between subsequent zeros for the first time.

Original entry on oeis.org

1, 31, 43, 61, 73, 95, 114, 146, 154, 161, 165, 199, 221, 233, 237, 246, 286, 330, 341, 354, 357, 359, 365, 374, 395, 402, 406, 410, 417, 421, 426, 443, 538, 586, 619, 665, 782, 787, 794, 797, 803, 813, 818, 830, 851, 861, 871, 879, 885, 887, 890, 894, 897, 901, 905, 907, 911

Offset: 1

M. F. Hasler, May 08 2018

nonn

This is related to the Mertens conjecture, more precisely to record values of Mertens function A002321 in the following sense: Due to the short-scale and long-scale oscillations of A002321, it is less appealing to consider record values in the usual sense (cf. A051402), which yields many slowly growing records and record indices lying closely together, during the approach of a "long-scale" record. Therefore this sequence considers maxima or minima between two subsequent zeros, ignoring the empty intervals between immediately adjacent zeros A002321(k) = A002321(k+1) = 0.
The values of these extrema are listed in A304240(n) = A002321(A304239(n)).
Then one can consider the sequence of indices where the corresponding values of A002321 have opposite sign, and/or are larger in absolute value than the preceding record amplitude in the above sense, cf. A304240 & A304241: These are the points which one would really consider as record maxima / minima when looking at the graph on a larger scale.

			The initial value a(1) = 1 may be considered conventional, or the maximum reached between M(0) = 0 (empty sum) and M(2) = 0, where we write M for the Mertens function A002321.
After M(2) = 0, Mertens's function has negative values up to the next zero, M(39) = 0. The largest negative value is -4 = M(31) = M(32). Therefore a(2) = 31.
Since M(39) = M(40) = 0, the maximum amplitude between these two consecutive zeros would be zero, and is ignored by definition.
The next "local minimum" of this type is reached at M(43) = -3, this value is taken several times up to the next zero at n = 58. Therefore a(3) = 43.
The next such "local minima" are M(61) = -2 and M(73) = -4, so a(4) = 61, a(5) = 73.
It is only at n = 94 that M takes a positive value for the first time after M(1) = 1, and M(95) = 2 is the largest value reached until the next zero (at n = 101), so a(6) = 95.
And so on.

Cf. A002321, A028442 (zeros of M), A051400, A051401, A051402 (where M, -M, |M| reaches k = 1, 2, 3, ...).

Cf. A304240 (values of the extrema), A304241 (indices of increasingly larger extrema), A304242 (the associated values).

M=0; for(n=1,oo, if(m=A002321(n),abs(m)>abs(M)&& [M,N]=[m,n], M&& M=printf(N",")))

A304241 Indices where Mertens function A002321 reaches record amplitudes between zeros.

Original entry on oeis.org

1, 31, 114, 199, 443, 665, 1109, 1637, 2803, 7021, 8511, 9861, 19291, 24185, 31990, 42961, 48433, 59577, 96014, 141869, 230399, 300551, 355733, 603151, 926265, 1066854, 1793918, 3239797, 5343761, 6481601, 7109110, 10194458, 12874814, 30919091, 61913863

Offset: 1

M. F. Hasler, May 08 2018

A subsequence of A051402 and A304239.

These are the indices where the Mertens function M = A002321 not only reaches a record value (in absolute value), but also its largest amplitude between subsequent zeros (as to avoid many "intermediate" records).

Bert Dobbelaere, Table of n, a(n) for n = 1..65
Bert Dobbelaere, C++ program for A304241 and A304242
G. Hurst, Computations of the Mertens Function and Improved Bounds on the Mertens Conjecture, arXiv:1610.08551 [math.NT], 2016-2017.

Cf. A002321, A028442 (zeros of M), A051400, A051401, A051402 (where M, -M, |M| reaches k = 1, 2, 3, ...).

Cf. A304242, A304239, A304240.

L=M=0;for(n=1,oo,if(m=A002321(n),abs(m)>abs(M)&&[M,N]=[m,n],abs(M)>abs(L) && (L=M) && print1(N",");M=0))

More terms from Bert Dobbelaere, Oct 30 2018

A304242 Increasingly larger (in absolute value) extrema of the Mertens function A002321 between subsequent zeros.

Original entry on oeis.org

1, -4, -6, -8, -9, -12, -15, -16, -25, -29, 35, -43, 51, -72, 73, -88, 96, -113, -132, -134, -154, 240, -258, -278, -368, 432, 550, -683, -847, 1060, -1078, 1240, -1447, -2573, 2845, -3448, -4610, -6226, 6695, -8565, 9132, 10246, -15335, -17334, 21777, -25071

Offset: 1

M. F. Hasler, May 08 2018

Values of A002321 at the indices listed in A304241.

These are those records of the absolute value of A002321 which are the maxima or minima between subsequent zeros. Figuratively speaking, these are the increasingly larger heights of the mountains or depths of the valleys of the graph of A002321.

Bert Dobbelaere, Table of n, a(n) for n = 1..65
Bert Dobbelaere, C++ program for A304241 and A304242
G. Hurst, Computations of the Mertens Function and Improved Bounds on the Mertens Conjecture, arXiv:1610.08551 [math.NT], 2016-2017.

Cf. A002321, A028442 (zeros of M), A051400, A051401, A051402 (where M, -M, |M| reaches k = 1, 2, 3, ...).

Cf. A304241, A304239, A304240.

L=M=0; for(n=1,oo, if(m=merten(n), abs(m)>abs(M) && [M,N]=[m,n], abs(M)>abs(L) && (L=M) && print1(L","); M=0))

print1(j=1);for(i=1,#A051402-1,while( A028442[j] < A051402[i], j++); if( A028442[j-(j>1)]<=A051402[i] && A028442[j] < A051402[i+1], print1(","A002321(A051402[i])))) \\ Using precomputed vectors A002321 and A051402, e.g. from the b-files: {c=0;AX=apply(t->fromdigits(digits(t)[#Str(c++)+1..-1]),readvec("/tmp/bX.txt"))}

a(n) = A002321(A304241(n)).

More terms from Bert Dobbelaere, Oct 30 2018

A171096 Solutions to the equation M(n) = -1 (M = Mertens's function A002321).

Original entry on oeis.org

3, 4, 6, 10, 15, 16, 22, 26, 27, 28, 35, 36, 38, 41, 57, 59, 60, 62, 63, 64, 66, 69, 87, 88, 91, 92, 102, 123, 124, 125, 126, 129, 134, 135, 136, 143, 144, 151, 152, 153, 155, 156, 158, 165, 167, 168, 169, 210, 213

Offset: 1

Neven Juric (neven.juric(AT)apis-it.hr), Sep 10 2010

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Mertens Function

Cf. A028442, A118684.

isok(n) = sum(k=1, n, moebius(k)) == -1; \\ Michel Marcus, Nov 20 2017

A304240 Extremum of the Mertens function A002321 between two successive (but not adjacent) zeros.

Original entry on oeis.org

1, -4, -3, -2, -4, 2, -6, 1, -2, 1, -1, -8, 5, -1, 1, -3, -8, 1, 3, -1, -1, -1, 1, -3, 2, -1, -1, -2, 2, -1, -1, -9, 1, 7, -5, -12, -1, -2, 1, -1, 3, 1, 3, -4, 1, -3, 2, 2, -1, -1, -1, -1, -1, 2, 1, -1, -1, 1, 1, 6, 1, 2, 1, -1, -15, -3, 1, -1, 2, 1, 2, -1, -1, 1, 1

Offset: 1

M. F. Hasler, May 08 2018

sign

In view of its definition, the Mertens function A002321 does not change sign between two successive zeros. Here we list the extrema, i.e., smallest or largest value, depending on the respective sign, between two zeros, excluding the case where these zeros are immediately adjacent, i.e., A002321(k) = A002321(k+1) = 0.

See A304239 and A304241 - A304242 for motivation & further information.

			The Mertens function M = A002321 is defined as partial sums of the Möbius function mu. At n = 1 it has the nonzero value M(1) = 1, and at n = 2 it has its first zero, M(2) = 0. Therefore we let a(1) = 1 by convention. (One can also consider that M(0) = 0, the empty sum, is an "initial zero" preceding M(1).)
Between the first and second zero of M = A002321, M(2) = 0 and M(39) = 0, M takes only negative values, and the largest in absolute value is a(2) = -4.
M(39) = 0 is immediately followed by another zero, M(40) = 0, the "empty" interval between these two is ignored by definition.
The next zero is at n = 58. Between n = 40 and n = 58 M takes only negative values, and the minimum is a(3) = -3.

Cf. A002321, A028442 (zeros of M), A051400, A051401, A051402 (where M, -M, |M| reaches k = 1, 2, 3, ...).

Cf. A304239, A304241, A304242.

M=0; for(n=1, oo, if(m=A002321(n), abs(m)>abs(M) && M=m, M && M=print1(M", ")))

a(n) = A002321(A304239(n)).

cp's OEIS Frontend

A057639 First differences of zero-sites (A028442) of Mertens's function A002321.

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A057627 Number of nonsquarefree numbers not exceeding n.

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A062982 Numbers n such that Mertens's function of n (A002321) is divisible by phi(n).

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A118684 Numbers k for which the Mertens function M(k) (A002321) is 1.

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A100766 Numbers for which the values of the Moebius function (A008683) and the Mertens function (A002321) are both 0.

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A304239 Indices for which the Mertens function A002321 reaches its extremum between subsequent zeros for the first time.

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A304241 Indices where Mertens function A002321 reaches record amplitudes between zeros.

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A304242 Increasingly larger (in absolute value) extrema of the Mertens function A002321 between subsequent zeros.