A002321 -id:A002321 - 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

A059581 From Von Sterneck's conjecture: floor(sqrt(n)) - 2*|Mertens's function A002321(n)|.

Original entry on oeis.org

-1, 1, -1, 0, -2, 0, -2, -2, -1, 1, -1, -1, -3, -1, 1, 2, 0, 0, -2, -2, 0, 2, 0, 0, 1, 3, 3, 3, 1, -1, -3, -3, -1, 1, 3, 4, 2, 4, 6, 6, 4, 2, 0, 0, 0, 2, 0, 0, 1, 1, 3, 3, 1, 1, 3, 3, 5, 7, 5, 5, 3, 5, 5, 6, 8, 6, 4, 4, 6, 4, 2, 2, 0, 2, 2, 2, 4, 2, 0, 0, 1, 3, 1, 1, 3, 5, 7, 7, 5, 5, 7, 7, 9

Offset: 1

N. J. A. Sloane, Feb 16 2001

sign

Von Sterneck conjectured that 2*|A002321(n)| < sqrt(n) for all sufficiently large n. This is now known to be false. This is different from the Mertens conjecture that |A002321(n)| < sqrt(n) for all n > 1 (which is also false).

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VI.2, p. 188.

Cf. A002321, A059571.

Table[Floor[Sqrt[n]] - 2 Abs[Plus @@ MoebiusMu[Range[n]]], {n, 1, 80}] (* Carl Najafi, Aug 17 2011 *)

A062984 a(n) = M(C(n)), where M(n) is Mertens's function (A002321) and C(n) is Chowla's function (A048050).

Original entry on oeis.org

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

Offset: 1

Jason Earls, Jul 25 2001

Harry J. Smith, Table of n, a(n) for n = 1..2000

Cf. A002321, A048050.

A062984[n_] := Sum[MoebiusMu[k], {k, DivisorSigma[1, n] - n - 1}];
Array[A062984, 100] (* Paolo Xausa, May 03 2024 *)

M(n)=sum(k=1,n,moebius(k));
C(n)=sigma(n)-n-1;
j=[]; for(n=1,350,j=concat(j,M(C(n)))); j

{ for (n=1, 2000, a=sum(k=1, sigma(n) - n - 1, moebius(k)); write("b062984.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 15 2009

A063473 a(n) = M(2*n-1), where M(n) is Mertens's function (A002321): Sum_{k=1..n} mu(k), where mu = Moebius function (A008683).

Original entry on oeis.org

1, -1, -2, -2, -2, -2, -3, -1, -2, -3, -2, -2, -2, -1, -2, -4, -3, -1, -2, 0, -1, -3, -3, -3, -3, -2, -3, -2, -1, -1, -2, -1, 0, -2, -1, -3, -4, -3, -2, -4, -4, -4, -3, -1, -2, -1, 0, 2, 1, 1, 0, -2, -3, -3, -4, -4, -5, -5, -5, -3, -3, -1, -1, -2, -1, -3, -2, -1, -2, -4, -3, -1, 0, 1, 0, -1, -1, -1, -2, 0, 1, 0, -1, -1, -1, -2, -3, -4, -3

Offset: 1

Jason Earls, Jul 27 2001

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A002321, A008683.

M(n)=sum(k=1,n,moebius(k));
j=[]; for(n=1,200,j=concat(j,M(2*n-1))); j

{ for (n=1, 1000, if (n>1, a+=moebius(2*n - 2) + moebius(2*n - 1), a=1); write("b063473.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 22 2009

A067195 Numbers n such that M(n) = Sum_{i=1..n} mu(sigma(i)) where M(n) is the Mertens function A002321(n).

Original entry on oeis.org

1, 2, 4, 30, 33, 38, 42, 45, 47, 48, 57, 59, 60, 64, 66, 69, 77, 82, 85, 104, 106, 118, 121, 194, 196, 201, 285, 287, 288, 290, 465, 467, 468, 470, 499, 500, 510, 610, 614, 626, 628, 631, 632, 642, 718, 723, 724, 785, 793, 798, 811, 812, 814, 823, 824, 825, 838

Offset: 1

Benoit Cloitre, Feb 19 2002

nonn

Indranil Ghosh, Table of n, a(n) for n = 1..263 (terms < 5000)

Cf. A000203, A008683.

M[n_] := Sum[MoebiusMu[k], {k, n}]; Select[Range@ 838, Sum[MoebiusMu[DivisorSigma[1, i]], {i, #}] == M[#] &] (* Indranil Ghosh, Mar 16 2017 *)

isok(n) = sum(i = 1, n, moebius(sigma(i))) == mertens(n); \\ Michel Marcus, Sep 24 2013

A067196 Numbers n such that M(n) = Sum_{i=1..n} mu(phi(i)) where M(n) is the Mertens function A002321(n).

Original entry on oeis.org

1, 6, 142, 154, 157, 167, 168, 169, 209, 213, 214, 231, 232, 235, 236, 238, 239, 240, 242, 243, 244, 245, 247, 248, 251, 252, 257, 259, 260, 261, 263, 264, 266, 269, 270, 278, 279, 280, 301, 318, 362, 363, 364, 366, 367, 368, 369, 371, 372, 391, 392, 402

Offset: 1

Benoit Cloitre, Feb 19 2002

nonn

mQ[n_]:=Sum[MoebiusMu[i],{i,n}]==Sum[MoebiusMu[EulerPhi[i]],{i,n}]; Select[Range[405],mQ[#] &] (* Jayanta Basu, May 22 2013 *)

A067265 Numbers n such that prime(n+1) - prime(n) = M(n) where M(n) is the Mertens function A002321(n).

Original entry on oeis.org

1, 226, 336, 395, 398, 552, 554, 583, 588, 805, 872, 926, 957, 961, 984, 995, 1008, 1263, 1275, 1363, 1384, 1443, 1447, 1450, 1456, 1462, 1946, 1949, 1957, 1964, 1988, 1991, 1992, 1997, 2008, 2023, 2028, 2037, 2055, 2076, 2107, 2175, 2203, 2234, 2240

Offset: 1

Benoit Cloitre, Feb 21 2002

Harvey P. Dale, Table of n, a(n) for n = 1..1000

PrimePi/@With[{nn=2500},Transpose[Select[Thread[{Partition[Prime[ Range[ nn+1]],2,1], Accumulate[Array[MoebiusMu,nn]]}]/.{{a_,b_},c_}-> {a,b,c}, #[[2]]-#[[1]]==#[[3]]&]][[1]]] (* Harvey P. Dale, Nov 23 2011 *)

A067266 Numbers n such that omega(n)=M(n) where omega(n) is A001221(n) and M(n) is the Mertens function A002321(n).

Original entry on oeis.org

95, 96, 97, 217, 228, 335, 337, 339, 342, 349, 395, 397, 398, 417, 543, 544, 546, 550, 603, 604, 605, 802, 804, 807, 808, 809, 817, 819, 820, 871, 872, 873, 879, 881, 901, 922, 930, 938, 945, 947, 949, 952, 962, 969, 971, 973, 975, 979, 981, 989, 991, 993

Offset: 1

Benoit Cloitre, Feb 21 2002

"omega(n)" (in the definition) means the number of prime factors of n counted without multiplicity, A001221. - Harvey P. Dale, Jul 14 2014

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1369 terms from Robert Israel)

Cf. A001221, A002321.

a067266 n = a067266_list !! (n-1)
a067266_list = filter (\x -> a001221 x == a002321 x) [1..]
-- Reinhard Zumkeller, Jul 14 2014

N:= 10^4: # to get all terms up to N
A:= [seq(numtheory[mobius](n),n=1..N)]:
Mertens:= map(round,Statistics:-CumulativeSum(A)):
omega:= t -> nops(numtheory:-factorset(t)):
select(t -> omega(t) = Mertens[t], [$1..N]); # Robert Israel, Jul 14 2014

With[{nn=1000},Flatten[Position[Thread[{Accumulate[Array[ MoebiusMu,nn]], PrimeNu[ Range[ nn]]}],?(First[#]==Last[#]&),{1},Heads->False]]] (* _Harvey P. Dale, Jul 14 2014 *)

isok(n) = (omega(n) == mertens(n)); \\ Michel Marcus, Sep 24 2013

A084236 a(n) = M(2^n), where M(n) is Mertens's function, A002321.

Original entry on oeis.org

1, 0, -1, -2, -1, -4, -1, -2, -1, -4, -4, 7, -19, 22, -32, 26, 14, -20, 24, -125, 257, -362, 228, -10, 211, -1042, 329, 330, -1703, 6222, -10374, 9569, 1814, -10339, -3421, 8435, 38176, -28118, 38729, -135944, 101597, 15295, -169338, 259886, -474483, 1726370, -3554573

Offset: 0

Robert G. Wilson v, May 15 2003

sign

Chai Wah Wu and Amiram Eldar, Table of n, a(n) for n = 0..75 (terms 0..73 from Hurst's paper added by Chai Wah Wu, terms 74..75 from Helfgott and Thompson's paper added by Amiram Eldar)
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, Mathematics of Computation, Vol. 87, No. 310 (2018), pp. 1013-1028; arXiv preprint, arXiv:1610.08551 [math.NT], 2016-2017.

Cf. A000079, A002321, A008683.

s = 0; i = 1; Do[ While[i <= 2^n, s = s + MoebiusMu[i]; i++ ]; Print[s], {n, 0, 50}]

a(n) = sum(k=1, 2^n, moebius(k)) \\ Indranil Ghosh, Mar 15 2017

a(n) = A002321(2^n).

a(n) = Sum_{k=1..2^n} mu(k), where mu = Moebius function (A008683).

a(31)-a(46) from Hurst's paper (copied by Charles R Greathouse IV, Oct 15 2018)

A127332 A126988 * A002321.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 5, 4, 4, 5, 9, 1, 10, 8, 3, 7, 15, 3, 16, 2, 6, 17, 21, -6, 13, 19, 11, 8, 27, -5, 27, 10, 13, 28, 10, -10, 35, 31, 17, -6, 40, -3, 40, 20, -4, 40, 44, -18, 32, 18, 26, 23, 50, 4, 21, 0, 28, 54, 58, -45, 59, 53, 3, 19, 24, 11, 65, 37, 39, 1

Offset: 1

Gary W. Adamson, Jan 10 2007

sign

			a(6) = 3 = 6*1 + 3*0 + 2*(-1) + 0*(-1) + 0*(-2) + 1*(-1), where (6, 3, 2, 0, 0, 1) = row 6 of A126988.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A002321, A054532, A054533, A054534, A054535, A126988.

Block[{nn = 70, m}, m = Table[Sum[MoebiusMu@k, {k, n}], {n, nn}]; Table[Total@ Array[m[[#]] If[Mod[n, #] == 0, n/#, 0] &, n], {n, nn}]] (* Michael De Vlieger, Jun 14 2018 *)

lista(nn) = {mat = matrix(nn, nn, n, k, if (n % k, 0, n/k)); vec = matrix(nn, 1, n, k, if (k==1, mertens(n), 0)); res = (mat*vec); for (n = 1, nn, print1(res[n, 1], ", "););} \\ Michel Marcus, Sep 25 2013

a(n) = sum(k=1, n, moebius(k / gcd(n, k)) * eulerphi(k) / eulerphi(k / gcd(n, k))); \\ Daniel Suteu, Jun 23 2018

M * V where M = A126988 as an infinite lower triangular matrix and V = the Mertens sequence, A002321 as a vector: [1, 0, -1, -1, -2, -1, ...].

a(n) = Sum_{q=1..n} c_q(n), where c_q(n) is the Ramanujan's sum function given in A054533. - Daniel Suteu, Jun 14 2018

Corrected and extended by Michel Marcus, Sep 25 2013

cp's OEIS Frontend

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

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A059581 From Von Sterneck's conjecture: floor(sqrt(n)) - 2*|Mertens's function A002321(n)|.

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A062984 a(n) = M(C(n)), where M(n) is Mertens's function (A002321) and C(n) is Chowla's function (A048050).

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A063473 a(n) = M(2*n-1), where M(n) is Mertens's function (A002321): Sum_{k=1..n} mu(k), where mu = Moebius function (A008683).

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A067195 Numbers n such that M(n) = Sum_{i=1..n} mu(sigma(i)) where M(n) is the Mertens function A002321(n).

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A067196 Numbers n such that M(n) = Sum_{i=1..n} mu(phi(i)) where M(n) is the Mertens function A002321(n).

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A067265 Numbers n such that prime(n+1) - prime(n) = M(n) where M(n) is the Mertens function A002321(n).

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A067266 Numbers n such that omega(n)=M(n) where omega(n) is A001221(n) and M(n) is the Mertens function A002321(n).

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A084236 a(n) = M(2^n), where M(n) is Mertens's function, A002321.

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