A051402 -id:A051402 - cp's OEIS frontend

A051400 Smallest value of x such that M(x) = n, where M() is Mertens's function A002321.

1, 95, 218, 219, 221, 554, 586, 1357, 1389, 1393, 1403, 1405, 1418, 3227, 3233, 3235, 3239, 3241, 3242, 3277, 3281, 3293, 3295, 8201, 8413, 8418, 8489, 8490, 8491, 8503, 8506, 8507, 8509, 8510, 8511, 11759, 11761, 11762, 11769, 11770, 11771, 11773

Offset: 1

Jud McCranie

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A002321, A051401, A051402.

a(n) = {x = 0; while (sum(k=1,x,moebius(k)) != n, x++); x} \\ Michel Marcus, Sep 24 2013

M(n)=sum(k=1,n,moebius(k));
print1(1,", "); c=M(1); n=2; while(n<10^4,if(M(n)>c,print1(n,", "); c=M(n)); n++) \\ Derek Orr, Jun 14 2016

A051401 Smallest value of x such that M(x) = -n, where M(x) is Mertens's function A002321.

Original entry on oeis.org

3, 5, 13, 31, 110, 114, 197, 199, 443, 659, 661, 665, 1105, 1106, 1109, 1637, 2769, 2770, 2778, 2791, 2794, 2795, 2797, 2802, 2803, 6986, 6987, 7013, 7021, 9717, 9718, 9719, 9721, 9726, 9741, 9749, 9822, 9823, 9830, 9831, 9833, 9857, 9861, 23833

Offset: 1

Jud McCranie

nonn

			M(31) = -4 and that is the first one, so a(4) = 31.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A002321, A008683, A051400, A051402.

s=0; t=0; Do[s=s+MoebiusMu[n]; If[sRobert G. Wilson v, Jul 03 2000 *)

A060434 An inverse to Mertens's function: smallest k >= 2 such that Mertens's function |M(k)| (see A002321) is equal to n.

Original entry on oeis.org

2, 3, 5, 13, 31, 110, 114, 197, 199, 443, 659, 661, 665, 1105, 1106, 1109, 1637, 2769, 2770, 2778, 2791, 2794, 2795, 2797, 2802, 2803, 6986, 6987, 7013, 7021, 8503, 8506, 8507, 8509, 8510, 8511, 9749, 9822, 9823, 9830, 9831, 9833, 9857, 9861, 19043

Offset: 0

Luis Rodriguez-Torres (ludovicusmagister(AT)yahoo.com), Apr 06 2001

Related to the Riemann Hypothesis through the Titchmarsh Theorem.

			M(1637) = 17 because the sum of Moebius mu(1) + mu(2) + ... + mu(1637) = 17.

Essentially same as A051402 except for initial terms.

with(numtheory): k := -1: s := 0: for n from 1 to 20000 do s := s+mobius(n): if (abs(s) > k) and (n>1) then k := abs(s): print(n, k); fi; od:

Reap[ For[ k = -1; s = 0; n = 1, n <= 20000, n++, s = s + MoebiusMu[n]; If[Abs[s] > k && n > 1, k = Abs[s]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Sep 04 2013, after Maple *)

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

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)).

A346457 a(n) is the smallest number k such that |Sum_{j=1..k} (-1)^omega(j)| = n, where omega(j) is the number of distinct primes dividing j.

Original entry on oeis.org

1, 4, 5, 8, 9, 32, 77, 88, 93, 94, 95, 96, 99, 100, 119, 124, 147, 148, 161, 162, 189, 206, 207, 208, 209, 210, 213, 214, 215, 216, 217, 218, 219, 226, 329, 330, 333, 334, 335, 394, 395, 416, 417, 424, 425, 428, 489, 514, 515, 544, 545, 546, 549, 554, 579, 584, 723, 724, 725, 800

Offset: 1

Ilya Gutkovskiy, Jul 19 2021

nonn

Robert Israel, Table of n, a(n) for n = 1..3223

Cf. A001221, A002053, A051400, A051401, A051402, A051470, A060434, A076479, A174863, A275547, A346455, A346456.

N:= 10000: # for values <= N
omega:= n -> nops(numtheory:-factorset(n)):
R:= map(n -> (-1)^omega(n),[$1..10000]):
S:= map(abs,ListTools:-PartialSums(R)):
m:= max(S):
V:= Vector(m):
for i from 1 to N do if S[i] > 0 and V[S[i]] = 0 then V[S[i]]:= i fi od:
convert(V,list); # Robert Israel, Oct 30 2023

Table[k=1;While[Abs[Sum[(-1)^PrimeNu@j,{j,k}]]!=n,k++];k,{n,30}] (* Giorgos Kalogeropoulos, Jul 19 2021 *)

a(n) = my(k=1); while (abs(sum(j=1, k, (-1)^omega(j))) != n, k++); k; \\ Michel Marcus, Jul 19 2021

a(n) = min {k : |Sum_{j=1..k} mu(rad(j))| = n}, where mu is the Moebius function and rad is the squarefree kernel.

A084234 Smallest k such that |M(k)| = n^2, where M(x) is Mertens's function A002321.

Original entry on oeis.org

1, 31, 443, 1637, 2803, 9749, 19111, 24110, 42833, 59426, 95514, 230227, 297335, 297573, 299129, 355541, 897531, 924717, 926173, 1062397, 1761649, 1763079, 1789062, 3214693, 3218010, 3232958, 4962865, 5307549, 5343710, 6433477, 6435874, 6473791, 9990083, 10188647

Offset: 1

Robert G. Wilson v, May 13 2003

nonn

"[I]f the absolute value of M(n) can be proved to be always less than the square root of n, then the Riemann Hypothesis is true. This is called Mertens's conjecture. ... Then along came Andrew Odlyzko and his colleague, Herman te Riele and they showed in 1984 that there is a number, far larger than 10^30, that invalidates Mertens's conjecture - call it N. In other words, M(N) is greater than the square of N. So the conjecture is not true." [Sabbagh]

Karl Sabbagh, The Riemann Hypothesis, The Greatest Unsolved Problem in Mathematics, Farrar, Straus and Giroux, New York, 2002, page 191.

Amiram Eldar, Table of n, a(n) for n = 1..100 (calculated using the b-file at A051402)

Cf. A002321, A051402.

i = s = 0; Do[While[Abs[s] < n^2, s = s + MoebiusMu[i]; i++ ]; Print[i - 1], {n, 1, 25}]

a(n) = A051402(n^2). - Amiram Eldar, May 06 2024

a(31)-a(34) from Amiram Eldar, May 06 2024

A346455 a(n) is the smallest number k such that Sum_{j=1..k} (-1)^omega(j) = n, where omega(j) is the number of distinct primes dividing j.

Original entry on oeis.org

1, 52, 55, 56, 57, 58, 77, 88, 93, 94, 95, 96, 99, 100, 119, 124, 147, 148, 161, 162, 189, 206, 207, 208, 209, 210, 213, 214, 215, 216, 217, 218, 219, 226, 329, 330, 333, 334, 335, 394, 395, 416, 417, 424, 425, 428, 489, 514, 515, 544, 545, 546, 549, 554, 579, 584, 723, 724, 725, 800

Offset: 1

Ilya Gutkovskiy, Jul 19 2021

nonn

Cf. A001221, A002053, A051400, A051401, A051402, A051470, A060434, A076479, A174863, A275547, A346456, A346457.

a[n_]:=(k=1;While[Sum[(-1)^PrimeNu@j,{j,k}]!=n,k++];k);Array[a,25] (* Giorgos Kalogeropoulos, Jul 19 2021 *)

a(n) = my(k=1); while (sum(j=1, k, (-1)^omega(j)) !=n, k++); k; \\ Michel Marcus, Jul 19 2021

a(n) = min {k : Sum_{j=1..k} mu(rad(j)) = n}, where mu is the Moebius function and rad is the squarefree kernel.

cp's OEIS Frontend

A051400 Smallest value of x such that M(x) = n, where M() is Mertens's function A002321.

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A051401 Smallest value of x such that M(x) = -n, where M(x) is Mertens's function A002321.

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A060434 An inverse to Mertens's function: smallest k >= 2 such that Mertens's function |M(k)| (see A002321) is equal to n.

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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.

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

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A346457 a(n) is the smallest number k such that |Sum_{j=1..k} (-1)^omega(j)| = n, where omega(j) is the number of distinct primes dividing j.

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