A025550 a(n) = ( 1/1 + 1/3 + 1/5 + ... + 1/(2*n-1) )*LCM(1, 3, 5, ..., 2*n-1).

1, 4, 23, 176, 563, 6508, 88069, 91072, 1593269, 31037876, 31730711, 744355888, 3788707301, 11552032628, 340028535787, 10686452707072, 10823198495797, 10952130239452, 409741429887649, 414022624965424, 17141894231615609, 743947082888833412, 750488463554668427

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy
• nice

Comments

Or, numerator of 1/1 + 1/3 + ... + 1/(2n-1) up to a(38).

Following similar remark by T. D. Noe in A025547, this coincides with f(n) = numerator of 1 + 1/3 + 1/5 + 1/7 + ... + 1/(2n-1) iff n <= 38. But a(39) = 18048708369314455836683437302413, f(39) = 1640791669937677803334857936583. Note that f(n) = numerator(digamma(n+1/2)/2 + log(2) + euler_gamma/2). - Paul Barry, Aug 19 2005 [See A350669(n-1).]

2*(1 + 1/3 + ... + 1/(2*n-1))/Pi = 2*a(n)/(A025547(n)*Pi) is the equivalent resistance between the points (0,0) and (n,n) on a 2-dimension infinite square grid of unit resistors. - Jianing Song, Apr 28 2025

Links

• Georg Fischer, Table of n, a(n) for n = 1..200 (first 39 terms from Jean-François Alcover)
• MathPages, The Algebra of an Infinite Grid of Resistors
• Physics Stack Exchange, On this infinite grid of resistors, what's the equivalent resistance?
• Eric Weisstein's World of Mathematics, Jeep Problem

Crossrefs

Cf. A025547, A075135, A002428, A350669.

Programs

Haskell

a025550 n = numerator $ sum  $ map (1 %) $ take n [1, 3 ..]
-- Reinhard Zumkeller, Jan 22 2012

Magma

[&+[1/d: d in i]*Lcm(i) where i is [1..2*n-1 by 2]: n in [1..21]]; // Bruno Berselli, Apr 16 2015

Maple

a:= n-> (f-> add(1/p, p=f)*ilcm(f[]))([2*i-1$i=1..n]):
seq(a(n), n=1..40);  # Alois P. Heinz, Apr 16 2015

Mathematica

Table[(Total[1/Range[1,2n-1,2]])LCM@@Range[1,2n-1,2],{n,30}] (* Harvey P. Dale, Sep 09 2020 *)

PARI

a(n)=my(v=vector(n,i,2*i-1));sum(i=1,#v,1/v[i])*lcm(v) \\ Charles R Greathouse IV, Feb 28 2013

Formula

1 + 1/3 + ... + 1/(2*n-1) = a(n)/A025547(n) = A350669(n-1)/A350670(n-1). - Jianing Song, Apr 28 2025

Extensions

Value of a(39) corrected by Jean-François Alcover, Apr 16 2015