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Search: a070888 -id:a070888
Displaying 1-5 of 5 results found. page 1
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A070891 Denominator of Sum_{k=1..n} mu(k)/k when it changes sign. +10
6
30, 15, 105, 210, 2310, 5005, 1616615, 9699690, 223092870, 111546435, 2156564410, 100280245065, 3710369067405, 7420738134810, 6541380665835015 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
Table of n, a(n) for n=1..15.
PROG
(PARI) t = 0; v = []; for( n = 1, 80, t1 = t; t = t + moebius( n) / n; if( t * t1 < 0, v = concat( v, denominator( t)), )); v
CROSSREFS
Cf. A070890, A070888, A070889.
KEYWORD
nonn,frac
AUTHOR
Donald S. McDonald, May 17 2002
STATUS
approved
A070889 Denominator of Sum_{k=1..n} mu(k)/k. +10
5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..2370
EXAMPLE
a(6) = 15 because 1 - 1/2 - 1/3 - 1/5 + 1/6 = 4/30 = 2/15.
MATHEMATICA
Table[ Denominator[ Sum[ MoebiusMu[k]/k, {k, 1, n}]], {n, 1, 37}]
PROG
(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
CROSSREFS
Cf. A008683, A068337, A070888 (numerators).
KEYWORD
frac,nonn
AUTHOR
Donald S. McDonald, May 17 2002
EXTENSIONS
Edited by Robert G. Wilson v, Jun 10 2002
STATUS
approved
A068337 a(n) = n!*Sum_{k=1..n} mu(k)/k, where mu(k) is the Möbius function. +10
4
1, 1, 1, 4, -4, 96, -48, -384, -3456, 328320, -17280, -207360, -481697280, -516741120, 79427174400, 1270834790400, 681401548800, 12265227878400, -6169334376038400, -123386687520768000, -158218429759488000, 47610136717000704000 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..450
Friedrich Roesler, Riemann's hypothesis as an eigenvalue problem, Linear Algebra and its Applications, Vol. 81 (1986), pp. 153-198.
Friedrich Roesler, Riemann's hypothesis as an eigenvalue problem. II, Linear Algebra and its Applications, Vol. 92 (1987), pp. 45-73.
FORMULA
a(n) = (-1)^(n-1)*{determinant of the n X n matrix m(i,j) = i+(j (mod i))} - Benoit Cloitre, May 28 2004
From Amiram Eldar, Oct 22 2020: (Start)
a(n) = A000142(n)*A070888(n)/A070889(n).
a(n) ~ O(n! * n^(-1/2 + eps)), for every eps>0, if and only if Riemann's hypothesis is true (Roesler, 1986). (End)
MATHEMATICA
n = 25; Accumulate[Table[MoebiusMu[k]/k, {k, 1, n}]] * Range[n]! (* Amiram Eldar, Oct 22 2020 *)
PROG
(Python)
from math import factorial
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 A068337(n): return factorial(n)*f(n) # Chai Wah Wu, Nov 03 2023
CROSSREFS
Cf. A000142, A008683, A070888, A070889.
KEYWORD
sign
AUTHOR
Leroy Quet, Feb 27 2002
STATUS
approved
A070890 Numerator of Sum_{k=1..n} mu(k)/k when it changes sign. +10
1
-1, 2, -1, 19, -1, 304, -81988, 410857, -249979, 4165258, -65721449, 2562470143, -5468849774, 184344882947, -137190436674212, 10026981687881, -12611493192339623, 519973962150962777, -8549627883788520181, 1874648830674470878723, -200643437220052588790575, 877316785444551755504875 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..22.
EXAMPLE
a(1)= -1 because numerator first sign change is 1-1/2-1/3-1/5= > (-1)/30 <0.
a(2)= 2 because next sign change is -1/30+1/6 = 2/15, reverts to positive.
PROG
(PARI) t = 0; v = []; for( n = 1, 120, t1 = t; t = t + moebius(n) / n; if( t * t1 < 0, v = concat( v, numerator( t)), )); v
CROSSREFS
Cf. A070891, A070888, A070889.
KEYWORD
frac,sign,easy
AUTHOR
Donald S. McDonald, May 17 2002
STATUS
approved
A070892 Numbers n such that absolute value of Sum_{k=1..n} mu(k)/k sets a new minimum. +10
0
1, 2, 3, 5, 7, 11, 61, 154, 857, 859, 2141, 2153, 2161, 39011, 39065, 39095, 56026, 56045, 56101, 56189, 56242, 56245, 56254, 56263, 56359, 2985634, 2985703, 2986715, 2986718, 2986721, 16904494, 16904497, 16904531, 16904534 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..34.
MATHEMATICA
s = 1; t = 0; Do[t = t + MoebiusMu[n] / n; If[ s > Abs[t], s = Abs[t]; Print[n]], {n, 1, 2 * 10^7}]
PROG
(PARI) t = 0.; t1 = 1; v = []; for( n = 1, 1400, t = t + moebius( n) / n; if( (t / t1 )^2 < 1, t1 = t; v = concat( v, n), )); v
CROSSREFS
Cf. A070888-A070891.
KEYWORD
nonn
AUTHOR
Donald S. McDonald, May 17 2002
EXTENSIONS
Edited and extended by Robert G. Wilson v, May 24 2002
STATUS
approved
page 1

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Last modified August 18 07:06 EDT 2024. Contains 375255 sequences. (Running on oeis4.)