A007406 -id:A007406 - cp's OEIS frontend

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

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

A128674 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 = 4.

Original entry on oeis.org

42, 110, 156, 272, 294, 342, 506, 812, 930, 1210, 1332, 1640, 1806, 2028, 2058, 2162, 2756, 3422, 3660, 4422, 4624, 4970, 5256, 6162, 6498, 6806, 7832, 9312, 10100, 10506, 11342, 11638, 11772, 12656, 13310, 14406, 16002, 17030, 18632, 19182, 22052, 22650, 23548, 24492, 26364

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 geometric progressions of the form (p-1)*p^k for k > 0 and some prime p > 5. Note the factorization of initial terms of {a(n)} = {6*7, 10*11, 12*13, 16*17, 6*7^2, 18*19, 22*23, 28*29, 30*31, 10*11*2, 36*37, 40*41, 42*43, 12*13^2, 6*7^3, ...}. 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, A125581, A126196, A126197, A128672, A128675, A128676.

k=4; 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} ]

Edited and extended by Max Alekseyev, May 09 2010

A103351 Numerators of sum_{k=1..n} 1/k^9 = Zeta(9,n).

Original entry on oeis.org

1, 513, 10097891, 5170139875, 10097934603139727, 373997614931101, 15092153145114981831307, 7727182467755471289426059, 4106541588424891370931874221019, 4106541592523201949266162797531

Offset: 1

Wolfdieter Lang, Feb 15 2005

a(n) gives the partial sums, Zeta(9,n), of Euler's Zeta(9). 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 A103352 and for the rationals Zeta(9,n) see the W. Lang link under A103345.

For k=1..8 see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052, A103345/A103346, A103347/A103348, A103349/A103350.

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

a(n) = numerator(sum_{k=1..n} 1/k^9).

G.f. for rationals Zeta(9, n): polylogarithm(9, x)/(1-x).

A120290 Numerator of generalized harmonic number H(p-1,2p)= Sum[ 1/k^(2p), {k,1,p-1}] divided by p^2 for prime p>3.

Original entry on oeis.org

2479157521, 159936660724017234488561, 1119583852472161859174156302552583713828739479026834819554843860744244189

Offset: 3

Alexander Adamchuk, Jul 08 2006

Generalized harmonic number is H(n,m)= Sum[ 1/k^m, {k,1,n} ]. The numerator of generalized harmonic number H(p-1,2p) is divisible by p^2 for prime p>3.

			With prime(3) = 5, a(3) = numerator[ 1 + 1/2^10 + 1/3^10 + 1/4^10 ] / 5^2 = 61978938025 / 25 = 2479157521.

Alexander Adamchuk, First 5 terms.
Eric Weisstein's World of Mathematics, Harmonic Number.
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem.

Cf. A119722, A099828, A099827, A001008, A007406, A007408, A007410.

Table[Numerator[Sum[1/k^(2*Prime[n]),{k,1,Prime[n]-1}]],{n,3,7}]/Table[Prime[n]^2,{n,3,7}]

a(n) = numerator[ Sum[ 1/k^(2*Prime[n]), {k,1,Prime[n]-1} ]] / Prime[n]^2 for n>2.

A128671 Least number k > 0 such that k^p does not divide the denominator of generalized harmonic number H(k,p) nor the denominator of alternating generalized harmonic number H'(k,p), where p = prime(n).

Original entry on oeis.org

20, 94556602, 444, 104, 77, 3504, 1107, 104, 2948, 903, 77, 1752, 77, 104, 370

Offset: 1

Alexander Adamchuk, Mar 24 2007, Mar 26 2007

Generalized harmonic numbers are defined as H(m,k) = Sum_{i=1..m} 1/i^k. Alternating generalized harmonic numbers are defined as H'(m,k) = Sum_{i=1..m} (-1)^(i+1)*1/i^k.
a(18)..a(24) = {77,104,77,136,104,370,136}. a(26)..a(27) = {77,104}.
a(n) is currently unknown for n = {16,17,25,...}. See more details in Comments at A128672 and A125581.

			a(2) = A128673(1) = 94556602.

Eric Weisstein's World of Mathematics, Harmonic Number

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

a(n) = A128670(prime(n)).

a(9) = 2948 and a(12) = 1752 from Max Alekseyev

Edited by Max Alekseyev, Feb 20 2019

A214508 Decimal expansion of the series limit sum_{k>=1} (-1)^(k+1) sum_{t=1..k} 1/(t^2*(k+1)^2).

Original entry on oeis.org

1, 6, 2, 6, 5, 4, 6, 6, 7, 3, 9, 7, 4, 2, 0, 0, 8, 0, 7, 7, 5, 5, 6, 4, 5, 6, 5, 1, 7, 3, 5, 9, 1, 1, 0, 1, 1, 8, 7, 0, 6, 4, 2, 0, 8, 3, 3, 7, 6, 5, 9, 9, 2, 3, 7, 2, 6, 7, 6, 3, 0, 6, 9, 8, 3, 1, 8, 4, 3, 5, 7, 7, 2, 9, 8, 2, 1, 0, 7, 4, 9, 2, 1, 6, 7, 2, 0, 0, 7, 4, 6, 3, 7, 5, 7, 4, 9, 8, 1, 0, 6, 7, 9, 6, 9

Offset: 0

R. J. Mathar, Jul 19 2012

Equals the alternating sum over (-1)^(k+1)*H_k^(2)/(k+1)^2, where H_k^(2) is the harmonic sum over inverse squares, H_k^(2) = sum_{t=1..k} 1/t^2 = 1, 5/4, 49/36, 205/144, 5269/3600,..., see A007406. The sum over H_k^(2)/(k+1)^2, over the absolute values, is Pi^4/120 = 0.811742425283353...

			0.162654667397420080...

D. H. Bailey, J. M. Borwein, R. Girgensohn, Experimental evaluation of Euler sums, Exp. Math. 3 (1994) 17, variable alpha(2,2)
G. Rutledge, R. D. Douglass, Table of definite integrals, Am. Math. Monthly 45 (1938) 525, variable A_4.

a099218 := polylog(4,1/2) ;
-4*a099218+13*Pi^4/288-7/2*Zeta(3)*log(2)+Pi^2/6*(log(2))^2-(log(2))^4/6 ;
evalf(%) ;

NSum[(-1)^(k + 1)*HarmonicNumber[k, 2]/(k + 1)^2, {k, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 110] // RealDigits[#, 10, 105] & // First (* or, from formula: *) 13*Pi^4/288 + 1/6*Pi^2*Log[2]^2 - 1/6*Log[2]*(Log[2]^3 + 21*Zeta[3]) - 4*PolyLog[4, 1/2] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Mar 06 2013 *)

13*Pi^4/288 + 1/6*Pi^2*log(2)^2 - 1/6*log(2)*(log(2)^3 + 21*zeta(3)) - 4*polylog(4, 1/2) \\ Charles R Greathouse IV, Jul 18 2014

Equals -4*A099218 +13*Pi^4/288 -7*A002117*log(2)/2+log^2(2)*(Pi^2-log^2(2))/6.

More terms from Jean-François Alcover, Feb 12 2013

A255008 Array T(n,k) read by ascending antidiagonals, where T(n,k) is the numerator of polygamma(n, 1) - polygamma(n, k).

Original entry on oeis.org

0, 0, -1, 0, 1, -3, 0, -2, 5, -11, 0, 6, -9, 49, -25, 0, -24, 51, -251, 205, -137, 0, 120, -99, 1393, -2035, 5269, -49, 0, -720, 975, -8051, 22369, -256103, 5369, -363, 0, 5040, -5805, 237245, -257875, 14001361, -28567, 266681, -761, 0, -40320

Offset: 0

Jean-François Alcover, Feb 12 2015

Up to signs, row n=0 is A001008/A002805, row n=1 is A007406/A007407 and column k=1 is n!.

			Array of fractions begin:
0,  -1,  -3/2,       -11/6,          -25/12,               -137/60, ...
0,   1,   5/4,       49/36,         205/144,             5269/3600, ...
0,  -2,  -9/4,    -251/108,       -2035/864,        -256103/108000, ...
0,   6,  51/8,    1393/216,      22369/3456,      14001361/2160000, ...
0, -24, -99/4,   -8051/324,   -257875/10368,   -806108207/32400000, ...
0, 120, 975/8, 237245/1944, 15187325/124416, 47463376609/388800000, ...
...

Eric Weisstein's MathWorld, Harmonic Number.
Eric Weisstein's MathWorld, Polygamma Function.
Wikipedia, Polygamma Function.

Cf. A001008, A002805, A007406, A007407, A255006, A255007, A255009 (denominators).

T[n_, k_] := (-1)^(n+1)*n!*HarmonicNumber[k-1, n+1] // Numerator; Table[T[n-k, k], {n, 0, 10}, {k, 1, n}] // Flatten

Fraction giving T(n,k) = polygamma(n, 1) - polygamma(n, k) = (-1)^(n+1)*n! * sum_{j=1..k-1} 1/j^(n+1) = (-1)^(n+1)*n!*H(k-1, n+1), where H(n,r) gives the n-th harmonic number of order r.

A255009 Array T(n,k) read by ascending antidiagonals, where T(n,k) is the denominator of polygamma(n, 1) - polygamma(n, k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 4, 6, 1, 1, 4, 36, 12, 1, 1, 8, 108, 144, 60, 1, 1, 4, 216, 864, 3600, 20, 1, 1, 8, 324, 3456, 108000, 3600, 140, 1, 1, 8, 1944, 10368, 2160000, 12000, 176400, 280, 1, 1, 16, 1944, 124416, 32400000, 2160000, 4116000

Offset: 0

Jean-François Alcover, Feb 12 2015

See A255008.

Cf. A001008, A002805, A007406, A007407, A255006, A255007, A255008 (numerators).

T[n_, k_] := (-1)^(n+1)*n!*HarmonicNumber[k-1, n+1] // Denominator; Table[T[n-k, k], {n, 0, 10}, {k, 1, n}] // Flatten

A276487 Denominator of Sum_{k=1..n} 1/k^n.

Original entry on oeis.org

1, 4, 216, 20736, 777600000, 46656000000, 768464444160000000, 247875891108249600000000, 4098310578334288576512000000000, 413109706296096288512409600000000, 7425496288284402957501110551810198732800000000000

Offset: 1

Ilya Gutkovskiy, Sep 05 2016

Also denominator of zeta(n) - Hurwitz zeta(n,n+1), where zeta(s) is the Riemann zeta function and Hurwitz zeta(s,a) is the Hurwitz zeta function.

Sum_{k>=1} 1/k^n = zeta(n).

			1, 5/4, 251/216, 22369/20736, 806108207/777600000, 47464376609/46656000000, 774879868932307123/768464444160000000, ...
a(3) = 216, because 1/1^3 + 1/2^3 + 1/3^3 = 251/216.

Eric Weisstein's World of Mathematics, Harmonic Number
Eric Weisstein's World of Mathematics, Riemann Zeta Function
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function

Cf. A001008, A002805, A007406, A007407, A031971, A276485 (numerators).

A276487:=n->denom(add(1/k^n, k=1..n)): seq(A276487(n), n=1..12); # Wesley Ivan Hurt, Sep 07 2016

Table[Denominator[HarmonicNumber[n, n]], {n, 1, 11}]

a(n) = denominator(sum(k=1, n, 1/k^n)); \\ Michel Marcus, Sep 06 2016

cp's OEIS Frontend

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

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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).

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A128674 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 = 4.

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A103351 Numerators of sum_{k=1..n} 1/k^9 = Zeta(9,n).

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A120290 Numerator of generalized harmonic number H(p-1,2p)= Sum[ 1/k^(2p), {k,1,p-1}] divided by p^2 for prime p>3.

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A128671 Least number k > 0 such that k^p does not divide the denominator of generalized harmonic number H(k,p) nor the denominator of alternating generalized harmonic number H'(k,p), where p = prime(n).

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A214508 Decimal expansion of the series limit sum_{k>=1} (-1)^(k+1) sum_{t=1..k} 1/(t^2*(k+1)^2).

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A255008 Array T(n,k) read by ascending antidiagonals, where T(n,k) is the numerator of polygamma(n, 1) - polygamma(n, k).

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