cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 152 results. Next

A028442 Numbers k such that Mertens's function M(k) (A002321) is zero.

Original entry on oeis.org

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
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Links

Crossrefs

Cf. A002321, A118684, A171096.

Programs

  • Magma
    M := func; [n: n in [1..500] | M(n) eq 0]; // Bruno Berselli, Jul 12 2021
  • Mathematica
    Select[Range[500], Plus@@MoebiusMu[Range[#]] == 0 &] (* Alonso del Arte, Jul 06 2004 *)
seq[kmax_] := Position[Accumulate[MoebiusMu[Range[kmax]]], 0] // Flatten; seq[500] (* Amiram Eldar, Jun 09 2024 *)
  • PARI
    isok(n) = sum(k=1, n, moebius(k)) == 0; \\ Michel Marcus, Nov 20 2017
  • PARI
    lista(kmax) = {my(s = 0); for(k = 1, kmax, s += moebius(k); if(s == 0, print1(k, ", ")));} \\ Amiram Eldar, Jun 09 2024
  • Perl
    use ntheory ":all"; for (moebius(1,1e7)) { ++$i; say $i unless $M+=$; } # _Dana Jacobsen, May 22 2015

A051402 Inverse Mertens function: smallest k such that |M(k)| = n, where M(x) is Mertens's function A002321.

Original entry on oeis.org

1, 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: 1

Views

Author

Jud McCranie

Keywords

Comments

From Torlach Rush, Oct 11 2018: (Start)
For k <= 10^7:
- a(n) is squarefree.
- if a(n) > M(k) then A008683(a(n)) is negative.
- if a(n) = M(k) then A008683(a(n)) is positive. (End)

Examples

			M(31) = -4, smallest one equal to +-4.

Links

Crossrefs

Cf. A002321, A008683, A051400, A051401.
Essentially same as A060434 except for initial terms.

Programs

  • Haskell
    import Data.List (elemIndex)
import Data.Maybe (fromJust)
a051402 = (+ 1) . fromJust . (`elemIndex` ms) where
   ms = map (abs . a002321) [1..]
-- Reinhard Zumkeller, Dec 26 2012
  • Maple
    with(numtheory): k := 0: s := 0: for n from 1 to 20000 do s := s+mobius(n): if abs(s) > k then k := abs(s): print(n); fi; od:
  • Mathematica
    a = s = 0; Do[s = s + MoebiusMu[n]; If[ Abs[s] > a, a = Abs[s]; Print[n]], {n, 1, 20000}]
  • PARI
    M(n)=sum(k=1,n,moebius(k));
print1(1,", "); c=M(1); n=2; while(n<10^3,if(abs(M(n))>c,print1(n,", "); c=abs(M(n))); n++) \\ Derek Orr, Jun 14 2016
  • PARI
    M(n) = sum(k=1, n, moebius(k));
a(n) = my(k = 1, s = moebius(1)); while (abs(s) != n, k++; s += moebius(k)); k; \\ Michel Marcus, Oct 12 2018

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

Original entry on oeis.org

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

Views

Author

Jud McCranie

Keywords

Links

Crossrefs

Cf. A002321, A051401, A051402.

Programs

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

Views

Author

Jud McCranie

Keywords

Examples

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

Links

Crossrefs

Cf. A002321, A008683, A051400, A051402.

Programs

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

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

Views

Author

Jason Earls, Jul 25 2001

Keywords

Comments

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

Links

Crossrefs

Cf. A002321.

Programs

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

Views

Author

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

Keywords

Examples

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

Links

Crossrefs

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.

Programs

  • Mathematica
    Select[Range[500], Plus @@ MoebiusMu[Range[#]] == 1 &] (* Carl Najafi, Aug 17 2011 *)
  • PARI
    isok(n) = sum(k=1, n, moebius(k)) == 1; \\ Michel Marcus, Nov 20 2017

A059571 From Mertens's conjecture (1): floor(sqrt(n)) - |M(n)|, where M is Mertens's function A002321.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 0, 1, 2, 1, 1, 0, 1, 2, 3, 2, 2, 1, 1, 2, 3, 2, 2, 3, 4, 4, 4, 3, 2, 1, 1, 2, 3, 4, 5, 4, 5, 6, 6, 5, 4, 3, 3, 3, 4, 3, 3, 4, 4, 5, 5, 4, 4, 5, 5, 6, 7, 6, 6, 5, 6, 6, 7, 8, 7, 6, 6, 7, 6, 5, 5, 4, 5, 5, 5, 6, 5, 4, 4, 5, 6, 5, 5, 6, 7, 8, 8, 7, 7, 8, 8, 9
Offset: 1

Views

Author

N. J. A. Sloane, Feb 16 2001

Keywords

Comments

Mertens conjectured that |A002321(n)| < sqrt(n) for all n > 1. This is now known to be false. So eventually there will be negative terms.

References

  • D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VI.2.
  • K. H. Rosen et al., eds., Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2000; p. 267.

Links

Crossrefs

Cf. A002321, A059572, A059581.

Programs

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

A100765 Numbers for which the values of the Moebius function (A008683) and the Mertens function (A002321) are both -1.

Original entry on oeis.org

3, 41, 59, 66, 102, 151, 165, 167, 233, 239, 255, 354, 357, 359, 367, 402, 406, 409, 421, 426, 429, 609, 638, 782, 786, 797, 826, 854, 885, 887, 890, 894, 897, 907, 911, 1015, 1019, 1221, 1259, 1281, 1283, 1298, 1301, 1303, 1307, 1319, 1327, 1493, 1526, 1533
Offset: 1

Views

Author

Alonso del Arte, Jan 03 2005

Keywords

Comments

This sequence is a subsequence of A100306, Numbers for which the values of the Moebius function and the Mertens function agree.

Examples

			102 is in the sequence because it is a sphenic number (exactly 3 distinct prime factors, A007304) number, so the Mobius function yields -1 and the sum of that value and the previous Mobius values (the Mertens function) is also -1.

Links

Crossrefs

Cf. A100306, A100766, A100767.

Programs

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

Extensions

Offset corrected by Donovan Johnson, Jun 19 2012

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

Views

Author

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

Keywords

Comments

Related to the Riemann Hypothesis through the Titchmarsh Theorem.

Examples

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

Crossrefs

Essentially same as A051402 except for initial terms.

Programs

  • Maple
    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:
  • Mathematica
    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 *)

A100306 Numbers for which the values of the Moebius function (A008683) and the Mertens function (A002321) agree.

Original entry on oeis.org

1, 3, 40, 41, 59, 66, 94, 102, 146, 150, 151, 160, 161, 164, 165, 167, 215, 232, 233, 236, 237, 239, 255, 330, 332, 333, 334, 354, 356, 357, 359, 363, 364, 365, 367, 394, 402, 404, 405, 406, 408, 409, 414, 415, 420, 421, 423, 424, 425, 426, 428, 429, 538, 542
Offset: 1

Views

Author

N. J. A. Sloane, Dec 30 2004

Keywords

Links

Crossrefs

Cf. A100765, A100766, A100767.

Programs

  • Maple
    with(numtheory): p:=proc(n) if mobius(n)=sum(mobius(k),k=1..n) then n else fi end: seq(p(n),n=1..700); # Emeric Deutsch, Feb 14 2005
  • Mathematica
    Select[Range[500], Plus @@ MoebiusMu[Range[#]] == MoebiusMu[#] &] (* Carl Najafi, Aug 17 2011 *)
  • PARI
    s=0; for(n=1,1e4, s+=t=moebius(n); if(t==s, print1(n", "))) \\ Charles R Greathouse IV, Jan 28 2014

Extensions

More terms from Emeric Deutsch, Feb 14 2005
Showing 1-10 of 152 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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