A007407 -id:A007407 - cp's OEIS frontend

A120077 Denominators of row sums of rational triangle A120072/A120073.

4, 36, 144, 3600, 3600, 176400, 705600, 6350400, 1270080, 153679680, 153679680, 25971865920, 25971865920, 129859329600, 519437318400, 150117385017600, 150117385017600, 54192375991353600, 2167695039654144, 1548353599752960, 221193371393280, 117011293467045120

Offset: 2

Wolfdieter Lang, Jul 20 2006

The first 19 terms coincide with A007407(n), for n>=2. However a(20) = 2167695039654144 and A007407(20) = 10838475198270720 = 5*a(20). Also a(21) = 1548353599752960 and A007407(21) = 221193371393280 = a(21)/7. From n = 22 up to at least n = 100 (checked) both sequences coincide again.
See the W. Lang link under A120072 for more details.
The corresponding numerators are given by A120076.
The n for which a(n) differs from A007407(n) are given by A309829. - Jeppe Stig Nielsen, Aug 18 2019

			The rationals A120076(m)/a(m), m>=2, begin with (3/4, 37/36, 169/144, 4549/3600, 4769/3600, ... ).

Jeppe Stig Nielsen, Table of n, a(n) for n = 2..1150

Cf. A007407, A120070, A120072, A120073, A120074, A120075, A120076, A309829.

A120077:= func< n | Denominator( (&+[1/k^2: k in [1..n]]) -1/n) >;
[A120077(n): n in [2..30]]; // G. C. Greubel, Apr 25 2023

Table[Denominator[HarmonicNumber[n,2] -1/n], {n,2,40}] (* G. C. Greubel, Apr 25 2023 *)

a(n) = denominator(sum(j=1,n-1,1/j^2-1/n^2)) \\ Jeppe Stig Nielsen, Aug 18 2019

a(n) = denominator(sum(j=1,n,1/j^2) - 1/n) \\ Jeppe Stig Nielsen, Aug 18 2019

def A120077(n): return denominator(harmonic_number(n,2) - 1/n)
[A120077(n) for n in range(2,31)] # G. C. Greubel, Apr 25 2023

a(n) = denominator(r(m)), with the rationals r(m) = Sum_{n=1..m-1} A120072(m,n)/A120073(m,n), m >= 2.
The rationals are r(m) = Zeta(2; m-1) - (m-1)/m^2, m>=2, with the partial sums Zeta(2; n) = Sum_{k=1..n} 1/k^2. See the W. Lang link under A103345.
O.g.f. for the rationals r(m), m>=2: log(1-x) + polylog(2,x)/(1-x).

a(21)-a(23) from Jeppe Stig Nielsen, Aug 18 2019

A128672 Numbers m such that m^k does not divide the denominator of the m-th generalized harmonic number H(m,k) nor the denominator of the m-th alternating generalized harmonic number H'(m,k), for k = 2.

Original entry on oeis.org

20, 42, 100, 110, 156, 272, 294, 342, 500, 506, 812, 930, 1210, 1332, 1640, 1806, 2028, 2058, 2162, 2500, 2756, 3422, 3660, 4422, 4624, 4970, 5256, 6162, 6498, 6806, 7832, 9312, 10100, 10506, 11026, 11342, 11638, 11772, 12500, 12656, 13310, 14406, 16002, 17030

Offset: 1

Alexander Adamchuk, Mar 20 2007

nonn

Generalized harmonic numbers are defined as H(m,k) = Sum_{j=1..m} 1/j^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{j=1..m} (-1)^(j+1)/j^k.

Sequence contains all geometric progressions of the form (p-1)*p^k for k > 0 and some primes p > 3. Note the factorization of initial terms of {a(n)} = {4*5, 6*7, 4*5^2, 10*11, 12*13, 16*17, 6*7^2, 18*19, 4*5^3, 22*23, 28*29, 30*31, 10*11^2, 36*37, 40*41, 42*43, 12*13^2, 6*7^3, 46*47, 4*5^4, 52*53, 58*59, 60*61, 66*67, 16*17^2, 70*71, 72*73, 78*79, 18*19^2, 82*83, ...}. The smallest term that does not fit this pattern is 11026 = ((149-1)/2) * 149.

Eric Weisstein's World of Mathematics, Harmonic Number

Similar sequences for generalized harmonic numbers with different k: A125581 (k=1), A128673 (k=3), A128674 (k=4), A128675 (k=5); A128676 (k=6).
For the least numbers k > 0 such that k^n does not divide the denominator of H(k,n) nor the denominator of H'(k,n), see A128670. See also A128671(n) = A128670(prime(n)).
Cf. A001008, A002805, A058313, A058312, A007406, A007407, A119682, A125581, A126196, A126197, A128674, A128675, A128676, A128673, A128670, A129671.

k=2; f=0; g=0; Do[ f=f+1/n^k; g=g+(-1)^(n+1)*1/n^k; kf=Denominator[f]; kg=Denominator[g]; If[ !IntegerQ[kf/n^k] && !IntegerQ[kg/n^k], Print[n] ], {n,1,7000} ]

Edited and extended by Max Alekseyev, May 07 2010

A128673 Numbers m such that m^k does not divide the denominator of the m-th generalized harmonic number H(m,k) nor the denominator of the m-th alternating generalized harmonic number H'(m,k), for k = 3.

Original entry on oeis.org

94556602, 141834903, 189113204, 283669806, 450820422

Offset: 1

Alexander Adamchuk, Apr 18 2007

Generalized harmonic numbers are defined as H(m,k) = Sum_{j=1..m} 1/j^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{j=1..m} (-1)^(j+1)/j^k.

Note that {a(n)} contains the following geometric progressions: ((16843-1)/3)*16843^m found by Max Alekseyev, ((16843-1)/2)*16843^m found by Max Alekseyev, ((16843-1)*2/3)*16843^m, (16843-1)*16843^m, 20826*21647^m found by Max Alekseyev, ((2124679-1)/3)*2124679^m, ((2124679-1)/2)*2124679^m, ((2124679-1)*2/3)*2124679^m, (2124679-1)*2124679^m. Here {16843, 2124679} = A088164 are the only two currently known Wolstenholme Primes: primes p such that {2p-1} choose {p-1} == 1 mod p^4. See more details in Comments at A128672 and A125581.

Eric Weisstein's World of Mathematics, Harmonic Number
Eric Weisstein's World of Mathematics, Wolstenholme Prime

Cf. A001008, A002805, A058313, A058312.
Cf. A007406, A007407, A119682, A007410, A120296, A099828.
Cf. A125581, A126196, A126197, A128672, A128674, A128675, A128676, A128670, A128671.
Cf. A088164 (Wolstenholme primes).

k=3; f=0; g=0; Do[ f=f+1/n^k; g=g+(-1)^(n+1)*1/n^k; kf=Denominator[f]; kg=Denominator[g]; If[ !IntegerQ[kf/n^k] && !IntegerQ[kg/n^k], Print[n] ], {n, 1, 450820422} ]

A128676 Numbers m such that m^k does not divide the denominator of the m-th generalized harmonic number H(m,k) nor the denominator of the m-th alternating generalized harmonic number H'(m,k), for k = 6.

Original entry on oeis.org

20, 100, 110, 156, 161, 272, 342, 345, 500, 506, 812, 930, 1210, 1332, 1640, 1806, 2028, 2162, 2500, 2756, 3051, 3422, 3660, 3703, 4422, 4624, 4970, 5256, 6162, 6498, 6806, 7832, 7935, 9312, 9605, 10100, 10506, 11342, 11638, 11772, 12500, 12656, 13310

Offset: 1

Alexander Adamchuk, Mar 20 2007

nonn

Generalized harmonic numbers are defined as H(m,k) = Sum_{j=1..m} 1/j^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{j=1..m} (-1)^(j+1)/j^k.

Sequence contains all terms of geometric progressions of the form (p-1)*p^k, k > 0, for some primes p >= 5, such as 4*5^k, 7*23^k, 15*23^k, 27*113^k, etc. Note the factorization of initial terms of {a(n)} = {4*5, 4*5^2, 10*11, 12*13, 7*23, 16*17, 18*19, 15*23, 4*5^3, 22*23, 28*29, 30*31, 10*11^2, 36*37, 40*41, 42*43, 12*13^2, 46*47, 4*5^4, 52*53, 27*113, 58*59, 60*61, 7*23^2, ...}. See more details in Comments at A128672 and A125581.

Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A001008, A002805, A058313, A058312.

Cf. A007406, A007407, A119682, A007410, A120296, A099828, A125581, A126196, A126197, A128672, A128673, A128676.

k=6; f=0; g=0; Do[ f=f+1/n^k; g=g+(-1)^(n+1)*1/n^k; kf=Denominator[f]; kg=Denominator[g]; If[ !IntegerQ[kf/n^k] && !IntegerQ[kg/n^k], Print[n] ], {n,1,3703} ]

Edited and extended by Max Alekseyev, May 08 2010

A163927 Numerators of the higher order exponential integral constants alpha(k,4).

Original entry on oeis.org

1, 49, 1897, 69553, 2515513, 90663937, 3264855049, 117543378001, 4231639039705, 152339702576545, 5484235568128681, 197432536935184369, 7107571838026381177, 255872590744254526273, 9211413307971174616393

Offset: 0

Johannes W. Meijer and Nico Baken, Aug 13 2009, Aug 17 2009

The higher order exponential integrals, see A163931, are defined by E(x,m,n) = x^(n-1)*Integral_{t>=x} E(t,m-1,n)/t^n for m >= 1 and n >= 1, with E(x,m=0,n) = exp(-x).
The series expansions of the higher order exponential integrals are dominated by the alpha(k,n) and the gamma(k,n) constants, see A090998.
The first Maple program uses the alpha(k,n) formula and the second the GF(z,n) to generate the alpha(k,n) coefficients in each column.
Appears to equal the numerator of the multiple harmonic (star) sum Sum_{1 <= k_1 <= ... <= k_n <= 3} 1/(k_1^2*...*k_n^2). If true, then a(n) = numerator( 3/2 - 3/(5*4^n) + 1/(10*9^n) ). - Peter Bala, Jan 31 2019

			a(k=0,n=4) = 1, a(k=1,4) = 49/36, a(k=2,4) = 1897/1296, a(k=3,4) = 69553/46656.

J. W. Meijer and N. H. G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No. 3, April 1987, pp. 209-211.

Cf. A163931 (E(x,m,n)), A090998 (gamma(k,n)).
Cf. A163928 y A163929.
a(k,1) = A000007(k)
a(k,2) = A000012(k) = 1^k.
a(k,3) = A002450(k+1)/A000302(k) with A000302(k) = 4^k.
a(k,4) = A163927(k)/A009980(k) with A009980(k) = 36^k.
The GF(z,n) lead to A008955.
The denominators of a(1,n), n >= 2, lead to A007407.

coln := 4; nmax := 15; kmax := nmax: k:=0: for n from 1 to nmax do alpha(k, n) := 1 od: for k from 1 to kmax do for n from 1 to nmax do alpha(k, n) := (1/k)*sum(sum(p^(-2*(k-i)), p=0..n-1)*alpha(i, n), i=0..k-1) od; od: seq(alpha(k, coln), k=0..nmax-1);
# End program 1
coln:=4; nmax1 := 16; for n from 0 to nmax1 do A008955(n, 0):=1 end do: for n from 0 to nmax1 do A008955(n, n) := (n!)^2 end do: for n from 1 to nmax1 do for m from 1 to n-1 do A008955(n, m) := A008955(n-1, m-1)*n^2 + A008955(n-1, m) end do: end do: m:=coln-1: f(m):=0: for n from 0 to m do f(m) := f(m) + (-1)^(n + m)*A008955(m, n)*z^(2*m-2*n) od: GF(z,coln) := m!^2/f(m): GF(z,coln):=series(GF(z,coln), z, nmax1);
# End program 2

alpha(k,n) = (1/k) * Sum_{i=0..k-1} (Sum_{p=0..n-1}(p^(2*i-2*k))*alpha(i, n)) with alpha(0,n) = 1, k >= 0 and n >= 1.
alpha(k,n) = alpha(k,n+1) -alpha(k-1,n+1)/n^2.
GF(z,n) = product((1-(z/k)^2)^(-1), k = 1..n-1) = (Pi*z/sin(Pi*z))/(Beta(n+z,n-z)/Beta(n,n)).

A103347 Numerators of Sum_{k=1..n} 1/k^7 = Zeta(7,n).

Original entry on oeis.org

1, 129, 282251, 36130315, 2822716691183, 940908897061, 774879868932307123, 99184670126682733619, 650750755630450535274259, 650750820166709327386387, 12681293156341501091194786541177, 12681293507322704937269896541177

Offset: 1

Wolfdieter Lang, Feb 15 2005

a(n) gives the partial sums, Zeta(7,n), of Euler's Zeta(7). Zeta(k,n) is also called H(k,n) because for k=1 these are the harmonic numbers H(n) A001008/A002805.

For the denominators see A103348 and for the rationals Zeta(7,n) see the W. Lang link under A103345.

Robert Israel, Table of n, a(n) for n = 1..336

For k=1..6 see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052, A103345/A103346.

f:= n -> numer(Psi(6,n+1)/720 + Zeta(7)):
map(f, [$1..20]); # Robert Israel, Mar 28 2018

s=0;lst={};Do[s+=n^1/n^8;AppendTo[lst,Numerator[s]],{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[ HarmonicNumber[n, 7] // Numerator, {n, 1, 12}] (* Jean-François Alcover, Dec 04 2013 *)

a(n) = numerator(sum_{k=1..n} 1/k^7).

G.f. for rationals Zeta(7, n): polylogarithm(7, x)/(1-x).

A027459 Numerator of Sum_{k=1..n} H(k)/k, where H(k) is k-th harmonic number.

Original entry on oeis.org

1, 7, 85, 415, 12019, 13489, 726301, 3144919, 30300391, 32160403, 4102360483, 4301068993, 758647585777, 112686856171, 3336876977, 96568406789, 28776062218037, 29608882035581, 1568274265798307, 11256448518043769

Offset: 1

Olivier Gérard

nonn

Originally defined as the first column of A027447, but now contains numerator in reduced form (cf. A329108). - Sean A. Irvine, Nov 04 2019
Numerators of the binomial transform of (-1)^n/(n+1)^3. The matrix a[i,j] below is the product of the binomial matrix and the matrix with general term binomial(i,j)(-1)^(i-j)/(i+1)^3. - Paul Barry, Aug 06 2004
From Alexander Adamchuk, Jan 02 2007 [edited by Jon E. Schoenfield, Mar 08 2015]: (Start)
Also a(n) is a numerator of S(n) = Sum_{k=1..n} H(k)/k, where H(k) is the k-th harmonic number, H(k) = Sum_{i=1..k} 1/i = A001008(k)/A002805(k).
S(n) = Sum_{k=1..n} H(k)/k = 1/2*(H(n)^2 + H(n,2)), where H(n,2) = Sum_{i=1..n} 1/i^2 = A007406(n)/A007407(n).
p divides a(p-1) and a(p-2) for prime p>3. a(n) is prime for n = {2, 7, 26, 31, 43, 53, 68, 80, 91, 123, 175, 236, 458, ...}. (End)
The n-fold repeated integral of (1/2)*log(x)^2 (all improper integrals with the lower limits of integration equal to 0) = x^n/n! * ( (1/2)*log(x)^2 - H(n)*log(x) + Sum_{k = 1..n} H(k)/k ). - Peter Bala, Feb 17 2022

			(a[ i,j ])^3 = MATRIX([[1, 0, 0, 0, 0], [7/8, 1/8, 0, 0, 0], [85/108, 19/108, 1/27, 0, 0], [415/576, 115/576, 37/576, 1/64, 0], [12019/18000, 3799/18000, 1489/18000, 61/2000, 1/125]]), n = 5.

Alexander Adamchuk, Table of n, a(n) for n = 1..30
Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A001008, A002805, A007406, A007407, A027447, A329108.

[Numerator(&+[HarmonicNumber(k)/k:k in [1..n]]):n in [1..20]]; // Marius A. Burtea, Nov 05 2019

Table[Numerator[Sum[Sum[1/i,{i,1,k}]/k,{k,1,n}]],{n,1,30}] (* Alexander Adamchuk, Jan 02 2007 *)
With[{nn=20},Accumulate[HarmonicNumber[Range[nn]]/Range[nn]]]//Numerator (* Harvey P. Dale, Feb 26 2023 *)

Numerators of sequence a(1, n) in (a(i, j))^3 where a(i, j) = 1/i if j <= i, 0 if j > i.
Numerators of (Wolstenholme(n, 1)^2 + Wolstenholme(n, 2))/(2*n)= ((gamma+Psi(n+1))^2 + Pi^2/6 - Psi(1, n+1))/(2*n), where Wolstenholme(n, m) = Sum_{i=1..n} 1/i^m. - Vladeta Jovovic, Aug 09 2002
a(n) = numerator(Sum_{k=1..n} ((Sum_{i=1..k} 1/i)/k)). - Alexander Adamchuk, Jan 02 2007

Corrected by Vladeta Jovovic, Aug 09 2002

A128675 Numbers m such that m^k does not divide the denominator of the m-th generalized harmonic number H(m,k) nor the denominator of the n-th alternating generalized harmonic number H'(m,k), for k = 5.

Original entry on oeis.org

444, 666, 888, 1332, 16428, 24642, 32856, 49284, 607836, 911754, 1215672, 1823508

Offset: 1

Alexander Adamchuk, Mar 20 2007

Generalized harmonic numbers are defined as H(m,k) = Sum_{j=1..m} 1/j^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{j=1..m} (-1)^(j+1)/j^k.

Sequence contains all terms of geometric progressions 37^k*(37-1)/3, 37^k*(37-1)/2, 37^k*(37-1)*2/3, 37^k*(37-1) for k > 0. Note the factorization of initial terms of {a(n)} = {37*12, 37*18, 37*24, 37*36, ...}. See more details in Comments at A128672 and A125581.

Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A001008, A002805, A058313, A058312, A007406, A007407, A119682, A007410, A120296, A099828, A125581, A126196, A126197, A128672, A128673, A128676.

k=5; f=0; g=0; Do[ f=f+1/n^k; g=g+(-1)^(n+1)*1/n^k; kf=Denominator[f]; kg=Denominator[g]; If[ !IntegerQ[kf/n^k] && !IntegerQ[kg/n^k], Print[n] ], {n,1,2000} ]

Eight more terms from Max Alekseyev, May 08 2010

A103349 Numerators of sum_{k=1..n} 1/k^8 = Zeta(8,n).

Original entry on oeis.org

1, 257, 1686433, 431733409, 168646292872321, 168646392872321, 972213062238348973121, 248886558707571775009601, 1632944749460578249437992161, 1632944765723715465050248417

Offset: 1

Wolfdieter Lang, Feb 15 2005

a(n) gives the partial sums, Zeta(8,n) of Euler's Zeta(8). Zeta(k,n) is also called H(k,n) because for k=1 these are the harmonic numbers H(n) A001008/A002805.

For the denominators see A103350 and for the rationals Zeta(8,n) see the W. Lang link under A103345.

For k=1..7 see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052, A103345/A103346, A103347/A103348.

s=0;lst={};Do[s+=n^1/n^9;AppendTo[lst,Numerator[s]],{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[ HarmonicNumber[n, 8] // Numerator, {n, 1, 10}] (* Jean-François Alcover, Dec 04 2013 *)
Accumulate[1/Range[10]^8]//Numerator (* Harvey P. Dale, Aug 11 2024 *)

a(n)=numerator(sum_{k=1..n} 1/k^8).

G.f. for rationals Zeta(8, n): polylogarithm(8, x)/(1-x).

A128670 Least number k > 0 such that k^n does not divide the denominator of generalized harmonic number H(k,n) nor the denominator of alternating generalized harmonic number H'(k,n).

Original entry on oeis.org

77, 20, 94556602, 42, 444, 20, 104, 42, 76, 20, 77, 110, 3504, 20, 903, 42, 1107, 20, 104, 42, 77, 20, 2948, 110, 136, 20, 76, 42, 903, 20, 77, 42, 268, 20, 7004, 110, 1752, 20, 19203, 42, 77, 20, 104, 42, 76, 20, 370, 110, 1107, 20, 77, 42, 12246, 20, 104, 42

Offset: 1

Alexander Adamchuk, Mar 24 2007

nonn

Generalized harmonic numbers are defined as H(m,k) = Sum_{j=1..m}1/j^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{j=1..m} (-1)^(j+1)/j^k.
Some apparent periodicity in {a(n)} (not without exceptions): a(n) = 20 for n = 2 + 4m, a(n) = 42 for n = 4 + 12m and 8 + 12m, a(n) = 76 for n = 9 + 18m, a(n) = 77 for n = 1 + 10m, a(n) = 104 for n = 7 + 12m, a(n) = 110 for n = 12m, a(n) = 136 for n = 25 + 32m, etc.
See more details in Comments at A128672 and A125581.

Max Alekseyev, Table of n, a(n) for n=1..158.
Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A001008, A002805, A058313, A058312, A007406, A007407, A119682, A007410, A120296, A125581, A126196, A126197, A128672, A128673, A128674, A128675, A128676, A128671, A128670.

More terms and b-file from Max Alekseyev, May 07 2010

cp's OEIS Frontend

A120077 Denominators of row sums of rational triangle A120072/A120073.

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A128672 Numbers m such that m^k does not divide the denominator of the m-th generalized harmonic number H(m,k) nor the denominator of the m-th alternating generalized harmonic number H'(m,k), for k = 2.

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A128673 Numbers m such that m^k does not divide the denominator of the m-th generalized harmonic number H(m,k) nor the denominator of the m-th alternating generalized harmonic number H'(m,k), for k = 3.

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A128676 Numbers m such that m^k does not divide the denominator of the m-th generalized harmonic number H(m,k) nor the denominator of the m-th alternating generalized harmonic number H'(m,k), for k = 6.

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A163927 Numerators of the higher order exponential integral constants alpha(k,4).

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A103347 Numerators of Sum_{k=1..n} 1/k^7 = Zeta(7,n).

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A027459 Numerator of Sum_{k=1..n} H(k)/k, where H(k) is k-th harmonic number.

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