Search: a070888 -id:a070888
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A070891
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Denominator of Sum_{k=1..n} mu(k)/k when it changes sign.
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+10
6
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30, 15, 105, 210, 2310, 5005, 1616615, 9699690, 223092870, 111546435, 2156564410, 100280245065, 3710369067405, 7420738134810, 6541380665835015
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OFFSET
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1,1
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LINKS
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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A070889
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Denominator of Sum_{k=1..n} mu(k)/k.
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+10
5
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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;
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OFFSET
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1,2
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LINKS
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EXAMPLE
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a(6) = 15 because 1 - 1/2 - 1/3 - 1/5 + 1/6 = 4/30 = 2/15.
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MATHEMATICA
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Table[ Denominator[ Sum[ MoebiusMu[k]/k, {k, 1, n}]], {n, 1, 37}]
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PROG
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(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)
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CROSSREFS
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KEYWORD
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frac,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A068337
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a(n) = n!*Sum_{k=1..n} mu(k)/k, where mu(k) is the Möbius function.
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+10
4
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1, 1, 1, 4, -4, 96, -48, -384, -3456, 328320, -17280, -207360, -481697280, -516741120, 79427174400, 1270834790400, 681401548800, 12265227878400, -6169334376038400, -123386687520768000, -158218429759488000, 47610136717000704000
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OFFSET
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1,4
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LINKS
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FORMULA
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a(n) = (-1)^(n-1)*{determinant of the n X n matrix m(i,j) = i+(j (mod i))} - Benoit Cloitre, May 28 2004
a(n) ~ O(n! * n^(-1/2 + eps)), for every eps>0, if and only if Riemann's hypothesis is true (Roesler, 1986). (End)
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MATHEMATICA
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n = 25; Accumulate[Table[MoebiusMu[k]/k, {k, 1, n}]] * Range[n]! (* Amiram Eldar, Oct 22 2020 *)
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PROG
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(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)
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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A070890
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Numerator of Sum_{k=1..n} mu(k)/k when it changes sign.
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+10
1
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-1, 2, -1, 19, -1, 304, -81988, 410857, -249979, 4165258, -65721449, 2562470143, -5468849774, 184344882947, -137190436674212, 10026981687881, -12611493192339623, 519973962150962777, -8549627883788520181, 1874648830674470878723, -200643437220052588790575, 877316785444551755504875
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graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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LINKS
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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KEYWORD
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frac,sign,easy
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AUTHOR
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STATUS
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approved
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A070892
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Numbers n such that absolute value of Sum_{k=1..n} mu(k)/k sets a new minimum.
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+10
0
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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)
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OFFSET
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1,2
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LINKS
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MATHEMATICA
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s = 1; t = 0; Do[t = t + MoebiusMu[n] / n; If[ s > Abs[t], s = Abs[t]; Print[n]], {n, 1, 2 * 10^7}]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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