A124432 -id:A124432 - cp's OEIS frontend

A096987 Numerator of Sum_{k=1..n} 1/H(k), where H(k) = Sum_{j=1..k} 1/j is the k-th harmonic number.

0, 1, 5, 73, 2221, 353777, 19595573, 239046803, 198972350083, 1535302297058707, 100536661265514127, 8974880059175708288297, 818810519369821323965929237, 990666575600755815615137883006341, 1220749860499992165560973207703210595953

Offset: 0

Leroy Quet, Aug 19 2004

			1/1 + 1/(1 + 1/2) + 1/(1 + 1/2 + 1/3) = 73/33, so a(3) = 73.

Cf. A124432 (denominators), A000720, A001008, A002805, A001620, A080130.

f[n_] := Numerator[ Sum[ 1/HarmonicNumber[j], {j, 1, n}]]; Table[ f[n], {n, 0, 14}] (* Robert G. Wilson v, Aug 21 2004 *)

m=13;for(n=0,m,print1(numerator(sum(k=1,n,1/sum(j=1,k,1/j))),",")) \\ Klaus Brockhaus, Aug 21 2004

From Thomas Ordowski, Mar 21 2023: (Start)
Sum_{k=1..n} 1/H(k) ~ Sum_{k=2..n} 1/log(k) ~ Integral_{2..n} dx/log(x) = Li(n).
Sum_{k=1..n} 1/H(k) = Sum_{k=1..n} 1/(log(k + 1/2) + gamma) - C + o(1), where gamma = A001620 = 0.577... is Euler's constant and the constant C = 0.0229825...
Sum_{k=1..n} 1/H(k) = exp(-gamma)*(Ei(log(n) + gamma) - 1) + o(1), where Ei(x) is the exponential integral function of real x, and we have Ei(log(x)) = li(x).
Note that a(n)/A124432(n) ~ pi(n) = A000720(n), see my first formula.
Sum_{k=1..n} 1/H(k) = n/(H(n) - 1 + ...) = n/(log(n) + gamma - 1 + O(1/log(n))).
Theorem: lim_{n->oo} (H(n) - n / Sum_{k=1..n} 1/H(k)) = 1, see my third formula.
Proof: since Integral dx / (log(x) + gamma) = exp(-gamma)*Ei(log(x) + gamma) + c, so we get lim_{n->oo} (log(n) + gamma - n*exp(gamma) / Ei(log(n) + gamma)) = 1, qed. (End)

More terms from Klaus Brockhaus and Robert G. Wilson v, Aug 21 2004

A229557 Array read by antidiagonals. Rows are the denominators of consecutive harmonic transforms starting with a first row 1, 1, 1,....

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 3, 1, 1, 1, 12, 33, 5, 1, 1, 1, 60, 825, 365, 8, 1, 1, 1, 20, 113025, 810665, 5992, 13, 1, 1, 1, 140, 5538225, 286794631705, 5886103384, 164541, 21, 1, 1, 1, 280, 60920475, 5619905141583441965, 4630449259971272605672, 14469935305431, 1031079, 34, 1, 1, 1

Offset: 1

Franz Vrabec, Sep 26 2013

The "harmonic transform" of a sequence of positive numbers a(i) is the sequence h(n) of the partial sums of their reciprocals: h(n)=sum_{i=1..n} 1/a(i).

			Table begins
1, 1,   1,      1,...
1, 1,   1,      1,...
1, 2,   6,     12,...
1, 3,  33,    825,...
1, 5, 365, 810665,...

Cf. A229556 (numerators).
Rows 1-4 are A000012(n), A000012(n), A002805(n), A124432(n+1).
Columns 1-2 are A000012(n), A000045(n+1).

A229556A := proc(n,k)
    option remember;
    if n = 1 then
        1;
    else
        add( 1/procname(n-1,c),c=1..k) ;
    end if;
end proc:
A229557 := proc(n,k)
    denom(A229556A(n,k)) ;
end proc:
for d from 2 to 12 do
    for k from d-1 to 1 by -1 do
        printf("%d,",A229557(d-k,k)) ;
    end do:
end do:

A234714 Numerator of sum_{k=1..n} 1/(k*H(k)) where H(k) is the harmonic number H(k) = sum_{j=1..n} 1/j.

Original entry on oeis.org

0, 1, 4, 50, 1349, 194713, 9917687, 112451057, 87707471002, 638247495586258, 39621419345255038, 3367553690081394959018, 293578866124447319211215128, 340463591070905769538621961175104, 403214792232827898020426758621769680732, 16787247654077861265551571547714793328259156

Offset: 0

Stuart Clary, Dec 29 2013

The corresponding denominators are in A234715.

Cf. A001008, A002805, A000254, A096987, A124432, A234715

nmax = 54; Table[ Numerator[ Sum[ 1/(k HarmonicNumber[k]), {k, 1, n} ] ], {n, 0, nmax} ]

A234715 Denominator of sum_{k=1..n} 1/(k*H(k)) where H(k) is the harmonic number H(k) = sum_{j=1..n} 1/j.

Original entry on oeis.org

1, 1, 3, 33, 825, 113025, 5538225, 60920475, 46360481475, 330503872435275, 20160736218551775, 1687675389591187637025, 145175524688023551724527525, 166370135063802174111446471957325, 194941377468714112878127508925972294225, 8038017817167489016303831575544615607779425

Offset: 0

Stuart Clary, Dec 29 2013

The corresponding numerators are in A234714.

A124432(n) = a(n) for 0 <= n <= 53, but A124432(54) = 3 * a(54).

Cf. A001008, A002805, A000254, A096987, A124432, A234714

nmax = 54; Table[ Denominator[ Sum[ 1/(k HarmonicNumber[k]), {k, 1, n} ] ], {n, 0, nmax} ]

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A096987 Numerator of Sum_{k=1..n} 1/H(k), where H(k) = Sum_{j=1..k} 1/j is the k-th harmonic number.

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A229557 Array read by antidiagonals. Rows are the denominators of consecutive harmonic transforms starting with a first row 1, 1, 1,....

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A234714 Numerator of sum_{k=1..n} 1/(k*H(k)) where H(k) is the harmonic number H(k) = sum_{j=1..n} 1/j.

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A234715 Denominator of sum_{k=1..n} 1/(k*H(k)) where H(k) is the harmonic number H(k) = sum_{j=1..n} 1/j.

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