A103933 -id:A103933 - cp's OEIS frontend

A103930 Numerators of squares of harmonic numbers A001008/A002805.

1, 9, 121, 625, 18769, 2401, 131769, 579121, 50822641, 54479161, 7007531521, 7399612441, 1313299956049, 1372958223289, 1429834803049, 5936819760481, 1775966959381729, 203755669038601, 75787776947048401, 3117562300468225

Offset: 1

Wolfdieter Lang, Mar 24 2005

The corresponding denominators are given in A103931.

W. Lang: Rationals H(n)^2.

A103930 := n->numer(sum(1/i, i=1..n)^2); seq(A103930(k), k=1..40); # Wesley Ivan Hurt, Sep 30 2013

a[n_] := HarmonicNumber[n]^2 // Numerator; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Sep 16 2013 *)

G.f.: -((d^3/dx^3)((log(1-x))^3))/3 + dilog(1-x)/(1-x) = ((log(1-x)^2) + dilog(1-x))/(1-x) with dilog(1-x)=polylog(2, x).
First differences give A103932(n)/A103933(n).
a(n) = numerator(H(n)^2), with the harmonic numbers H(n) = A001008(n)/A002805(n), n >= 1.

A103932 Numerators of first difference of squares of harmonic numbers.

Original entry on oeis.org

1, 5, 10, 47, 131, 71, 353, 1487, 6989, 1451, 82451, 42433, 1132133, 1158863, 236749, 4828073, 41781863, 42482563, 273253759, 277235737, 56204647, 18975625, 441730115, 670193263, 33874048171, 34224132367, 311048966203, 313970420453

Offset: 1

Wolfdieter Lang, Mar 24 2005

The corresponding denominators are given in A103933.

h(n+1) + h(n) = (n+1)*(h(n+1)^2 - h(n)^2), where h(n) is the n-th harmonic number. - Gary Detlefs, May 25 2012

Robert Israel, Table of n, a(n) for n = 1..2296
Wolfdieter Lang, Rationals.

H:= Vector(51):
for i from 2 to 51 do H[i]:= H[i-1]+1/(i-1) od:
HS:= map(t -> t^2, H):
convert(map(numer, HS[2..-1]-HS[1..-2]),list); # Robert Israel, Sep 27 2023

Array[ HarmonicNumber[#]^2&, 29, 0] // Differences // Numerator (* Jean-François Alcover, Jul 09 2013 *)

a(n) = numerator(r(n)), with the rationals r(n) = H(n)^2 - H(n-1)^2 where H(n) = A001008(n)/A002805(n), n >= 1, H(0):=0.
G.f. for r(n): (log(1-x))^2 + dilog(1-x) where dilog(1-x) = polylog(2, x).
a(n) = numerator(h(n) + h(n-1)), where h(n) is the n-th harmonic number. - Gary Detlefs, May 25 2012

cp's OEIS Frontend

A103930 Numerators of squares of harmonic numbers A001008/A002805.

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