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 1-3 of 3 results.

A070888 Numerator of Sum_{k=1..n} mu(k)/k.

Original entry on oeis.org

1, 1, 1, 1, -1, 2, -1, -1, -1, 19, -1, -1, -2323, -89, 304, 304, 163, 163, -81988, -81988, -15019, 410857, -249979, -249979, -249979, 4165258, 4165258, 4165258, 9246047, -65721449, -4193929329, -4193929329, -6504197377, -302679716, 2562470143
Offset: 1

Views

Author

Donald S. McDonald, May 17 2002

Keywords

Comments

Sum_{k>0} mu(k)/k = limit_{n->oo} A070888(n)/A070889(n) = 0. - Jean-François Alcover, Apr 18 2013. This is equivalent to the Prime Number Theorem! - N. J. A. Sloane, Feb 04 2022

Examples

			a(6) = 2 because 1-1/2-1/3-1/5+1/6 = 4/30 = 2/15.

References

  • Harold M. Edwards, Riemann's Zeta Function, Dover Publications, New York, 1974 (ISBN 978-0-486-41740-0), p. 92.
  • Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing, NY 1953, p. 568.

Links

Crossrefs

Cf. A008683, A068337, A070889 (denominators).

Programs

  • Maple
    T:= 0:
for n from 1 to 100 do
  T:= T + numtheory:-mobius(n)/n;
  A[n]:= numer(T)
od:
seq(A[n],n=1..100); # Robert Israel, Aug 04 2014
  • Mathematica
    Table[ Numerator[ Sum[ MoebiusMu[k]/k, {k, 1, n}]], {n, 1, 37}]
  • PARI
    t = 0; v = []; for( n = 1, 60, t= t + moebius( n) / n; v = concat( v, numerator( t))); v \\ adapted to latest PARI version by Michel Marcus, Aug 04 2014
  • Python
    from functools import lru_cache
from sympy import harmonic
@lru_cache(maxsize=None)
def f(n):
    if n <= 1:
        return 1
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (harmonic(j-1)-harmonic(j2-1))*f(k1)
        j, k1 = j2, n//j2
    return c+harmonic(j-1)-harmonic(n)
def A070888(n): return f(n).numerator # Chai Wah Wu, Nov 03 2023

Extensions

Edited by Robert G. Wilson v, Jun 10 2002

A070889 Denominator of Sum_{k=1..n} mu(k)/k.

Original entry on oeis.org

1, 2, 6, 6, 30, 15, 105, 105, 105, 210, 2310, 2310, 30030, 15015, 5005, 5005, 85085, 85085, 1616615, 1616615, 4849845, 9699690, 223092870, 223092870, 223092870, 111546435, 111546435, 111546435, 3234846615, 2156564410, 66853496710
Offset: 1

Views

Author

Donald S. McDonald, May 17 2002

Keywords

Examples

			a(6) = 15 because 1 - 1/2 - 1/3 - 1/5 + 1/6 = 4/30 = 2/15.

Links

Crossrefs

Cf. A008683, A068337, A070888 (numerators).

Programs

  • Mathematica
    Table[ Denominator[ Sum[ MoebiusMu[k]/k, {k, 1, n}]], {n, 1, 37}]
  • PARI
    t = 0; v = []; for( n = 1, 30, t = t + moebius( n) / n; v = concat( v, denominator( t))); v
  • Python
    from functools import lru_cache
from sympy import harmonic
@lru_cache(maxsize=None)
def f(n):
    if n <= 1:
        return 1
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (harmonic(j-1)-harmonic(j2-1))*f(k1)
        j, k1 = j2, n//j2
    return c+harmonic(j-1)-harmonic(n)
def A070889(n): return f(n).denominator # Chai Wah Wu, Nov 03 2023

Extensions

Edited by Robert G. Wilson v, Jun 10 2002

A266378 Recursive 2-parameter sequence allowing calculation of the Möbius function.

Original entry on oeis.org

-1, -1, 0, -1, -1, -1, -1, -1, -1, 0, 1, 2, 2, 1, -1, -3, -6, -10, -14, -17, -17, -14, -9, -4, -1, -1, -3, -5, -6, -5, -2, 3, 9, 14, 16, 14, 9, 4, 1, 0, 0, -1, -4, -9, -15, -20, -22, -19, -10, 5, 24, 43, 58, 67, 70, 67, 58, 44, 28, 14, 5, 1, 0, -1, -5, -14, -29, -49, -71, -90, -100, -95, -70, -23, 44, 126, 216, 305, 382, 436, 459, 449, 411
Offset: 3

Views

Author

Gevorg Hmayakyan, Dec 28 2015

Keywords

Comments

The a(n,m) forms a table where each row has (n-2)*(n-3)/2+1 = A000124(n-3) elements.
The index of the first row is n=3 and the index of the first column is m=0.
The right diagonal a(n, A000217(n-3)) = A008683(n), Möbius numbers, for n>=3.

Examples

			The first few rows are
-1;
-1,  0;
-1, -1, -1,  -1;
-1, -1,  0,   1,   2,   2,   1;
-1, -3, -6, -10, -14, -17, -17, -14, -9, -4, -1;

Links

Crossrefs

Cf. A002321, A008302, A008683, A068337.

Programs

  • Maple
    T := proc(n, k) option remember; if k=0 then 1 else if (n=1 and k=1) then 0 else if (k<0 or k>binomial(n, 2)) then 0 else T(n-1, k)+T(n, k-1)-T(n-1, k-n) end if end if end if end proc:
nu := n->((n-2)*(n-3))/(2):
a := proc (n, m) option remember; if m = 0 then -1 elif m < 0 or nu(n) < m then 0 else procname(n, m-1)+procname(n-1, m)-procname(n-1, m-n+1)-procname(n-1, nu(n-1))*(T(n-2, m-1)-T(n-2, m-2)) end if end proc:
L := seq(seq(a(N, r), r = 0 .. nu(N)), N = 3 .. 20);

Formula

a(n,m) = a(n, m-1)+a(n-1, m)-a(n-1, m-n+1)-a(n-1, nu(n-1))*(T(n-2, m-1)-T(n-2, m-2)), where T(n,m) are coefficients of A008302, nu(n)=(n-2)*(n-3)/2, a(n,0)=-1, a(n,m)=0 if m<0 and m>nu(n).
Möbius(n) = a(n,nu(n)).
Easy to prove:
a(n+1,1)-a(n,1) = -1 - Möbius(n), n>=3 and accordingly
a(n,1) = 3 - n - A002321(n-1).
a(n+1,2)-a(n,2) = 2 - n - A002321(n) - Möbius(n)*(n-3), n>3 and accordingly
a(n,2) = -1 - (n-1)*(n-4)/2 - (n-3)*A002321(n-1).
a(n,-1+(n-2)*(n-3)/2) = Möbius(n+1) + Möbius(n)*(n-3).
a(n,-2+(n-2)*(n-3)/2) = Möbius(n+2) + Möbius(n+1)*(n-3) + (1/2)*Möbius(n)*(n-1)*(n-4) and in general:
a(n,-k+(n-2)*(n-3)/2)=f(n,k), for n>k.
a(n,-k-n+(n-2)*(n-3)/2)=f(n,n+k) + f(n,k) + h(n, k), for n>k,
where:
f(n,k)=Sum_{m=0..k}{Möbius(n+k-m)*(T(n-1,m)-T(n-1,m-1))}
h(n,s)=Sum_{k=1..n}{Sum_{i=0..k+1)}{(-1)^(i+1)*binomial(k+1, i)*f(n+k, s-2*k+1-i)}}
Easy to see:
a(n,2)-(n-3)*a(n,1)=(n-3)*(n-4)/2
Conjecture:
Sum_{m=0..(n-2)*(n-3)/2}{a(n,m)} = -A068337(n-1).
Sum_{m=0..(n-2)*(n-3)/2}{a(n,m)*m^k*(-1)^m} = 0, n>2*k+4.
Showing 1-3 of 3 results.

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