%I #20 Nov 20 2022 01:51:51

%S 0,1,1,1,1,5,1,7,1,3,5,11,1,13,7,5,2,17,3,19,5,7,11,23,1,25,13,9,7,29,

%T 5,31,4,11,17,35,3,37,19,13,5,41,7,43,11,15,23,47,2,49,25,17,13,53,9,

%U 55,7,19,29,59,5,61,31,21,8,65,11,67,17,23,35,71,3,73

%N Numerator of b(n)-b(n+1), where b(n) = n/((n+1)(n+2)) = A026741/A045896.

%H Amiram Eldar, <a href="/A051712/b051712.txt">Table of n, a(n) for n = 1..10000</a>

%H Masanobu Kaneko, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/KANEKO/AT-kaneko.html">The Akiyama-Tanigawa algorithm for Bernoulli numbers</a>, J. Integer Sequences, 3 (2000), Article 00.2.9.

%F c(n) = a(n+1) is multiplicative with c(2^e) = 2^(e-3) if e > 2 and 1 otherwise, c(3^e) = 3^(e-1), and c(p^e) = p^e if p >= 5. [corrected by _Amiram Eldar_, Nov 20 2022]

%F Sum_{k=1..n} a(k) ~ (301/1152) * n^2. - _Amiram Eldar_, Nov 20 2022

%e 0, 1/60, 1/60, 1/70, 1/84, 5/504, 1/120, 7/990, 1/165, 3/572,...

%t b[n_] := n/((n + 1) (n + 2)); Numerator[-Differences[Array[b, 100]]]

%t (* or *)

%t f[p_, e_] := p^e; f[2, e_] := If[e < 3, 1, 2^(e - 3)]; f[3, e_] := 3^(e - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n - 1]; Array[a, 100] (* _Amiram Eldar_, Nov 20 2022 *)

%Y Cf. A026741, A045896, A051713.

%Y Row 3 of table in A051714/A051715.

%K nonn,frac,easy,mult

%O 1,6

%A _N. J. A. Sloane_