A119248 a(n) is the difference between denominator and numerator of the n-th alternating harmonic number Sum_{k=1..n} (-1)^(k+1)/k = A058313(n)/A058312(n).

0, 1, 1, 5, 13, 23, 101, 307, 641, 893, 7303, 9613, 97249, 122989, 19793, 48595, 681971, 818107, 13093585, 77107553, 66022193, 76603673, 1529091919, 1752184789, 7690078169, 8719737569, 23184641107, 3721854001, 96460418429

Offset: 1

Author

Alexander Adamchuk, Jul 22 2006

Keywords

• nonn
• easy

Comments

a(n)/A058312(n) = 1 - A058313(n)/A058312(n) appears in the locker puzzle (see the links in A364317) for the probability of success with the strategy used there for n lockers and allowed openings of up to floor(n/2) lockers. Note that gcd(a(n), A058312(n)) = 1. - Wolfdieter Lang, Aug 12 2023

Crossrefs

Cf. A058312, A058313, A075829, A364317.

Programs

Mathematica

Denominator[Table[Sum[(-1)^(k+1)/k,{k,1,n}],{n,1,30}]]-Numerator[Table[Sum[(-1)^(k+1)/k,{k,1,n}],{n,1,30}]]

PARI

a(n) = my(x=sum(k=1, n, (-1)^(k+1)/k)); denominator(x) - numerator(x); \\ Michel Marcus, May 07 2020

Formula

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/k) - numerator(Sum_{k=1..n} (-1)^(k+1)/k).

a(n) = A058312(n) - A058313(n).

a(n) = A075829(n+1).

a(n) = numerator(Sum_{k=2..n} (-1)^k/k). (Cf. A024168.) - Petros Hadjicostas, May 17 2020