A007407 -id:A007407 - cp's OEIS frontend

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.

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).

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

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

A373440 Denominator of sum of reciprocals of square divisors of n.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 4, 1, 1, 1, 16, 1, 9, 1, 4, 1, 1, 1, 4, 25, 1, 9, 4, 1, 1, 1, 16, 1, 1, 1, 18, 1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 16, 49, 25, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 64, 1, 1, 1, 4, 1, 1, 1, 18, 1, 1, 25, 4, 1, 1, 1, 16, 81, 1, 1, 4, 1

Offset: 1

Ilya Gutkovskiy, Jun 05 2024

			1, 1, 1, 5/4, 1, 1, 1, 5/4, 10/9, 1, 1, 5/4, 1, 1, 1, 21/16, 1, 10/9, 1, 5/4, 1, 1, 1, 5/4, 26/25, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007407, A007947, A017666, A017668, A035316, A332881, A373439 (numerators).

nmax = 85; CoefficientList[Series[Sum[x^(k^2)/(k^2 (1 - x^(k^2))), {k, 1, nmax}], {x, 0, nmax}], x] // Rest // Denominator
f[p_, e_] := (p^2 - p^(-2*Floor[e/2]))/(p^2-1); a[1] = 1; a[n_] := Denominator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Jun 26 2024 *)

a(n) = denominator(sumdiv(n, d, if (issquare(d), 1/d))); \\ Michel Marcus, Jun 05 2024

Denominators of coefficients in expansion of Sum_{k>=1} x^(k^2)/(k^2*(1-x^(k^2))).

a(n) is the denominator of Sum_{d^2|n} 1/d^2.

A035166 Let d(m) = denominator of Sum_{k=1..m} 1/k^2 and consider f(m) = product of primes which appear to odd powers in d(m); sequence lists m such that f(m) is different from f(m-1).

Original entry on oeis.org

1, 10, 15, 20, 25, 42, 49, 50, 55, 66, 75, 78, 91, 100, 110, 121, 125, 136, 153, 156, 164, 169, 171, 182, 189, 190, 205, 250, 253, 272, 276, 289, 294, 342, 343, 354, 361, 375, 406, 413, 435, 465, 473, 496, 500, 506, 516, 529, 555, 592, 605, 625

Offset: 1

Bill Gosper, Sep 04 2002

The prime 479 first appears in f(m) at m = 2395, ahead of 71, which first appears in f(2485).
The first occurrence of four distinct primes is at m = 2500, with 5^7, 17^3, 71 and 479.
For 1890 < m < 2006, d(m) is a square (f(m)=1). The lone prime in 1875 .. 1890 is 61 and in 2006 .. 2027 it is 59.
It appears that f(m) can differ from f(m-1) in at most one prime.
(f from definition) = A007913, squarefree part. - Reinhard Zumkeller, Jul 06 2012

			f(10) = 5 is the first time f(m) > 1. The 5 persists until it disappears at m = 15.

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

Cf. A075326, A075327, A007407.

import Data.List (findIndices)
a035166 n = a035166_list !! (n-1)
a035166_list = map (+ 1) $ findIndices (/= 0) $ zipWith (-) (tail gs) gs
   where gs = 0 : map a007913 a007407_list
-- Reinhard Zumkeller, Jul 06 2012

for k:1 do (subset(factor_number(denom(harmonic(k,2))), lambda([x],oddp(second(x)))), if old#old:%% then print(k,%%))

d[n_] := Denominator[ HarmonicNumber[n, 2]]; f[n_] := Times @@ Select[ FactorInteger[d[n]], OddQ[#[[2]]]&][[All, 1]]; A035166 = Join[{1}, Select[ Range[1000], f[#] != f[#-1]&]] (* Jean-François Alcover, Feb 26 2016 *)

d(m) = denominator(sum(k=1, m, 1/k^2));
f(m) = my(f=factor(d(m))); for (k=1, #f~, if (!(f[k,2] % 2), f[k,2] = 0)); factorback(f);
isok(m) = if (m==1, 1, f(m) != f(m-1)); \\ Michel Marcus, Sep 06 2023

A070985 Number of terms in the simple continued fraction for Sum_{k=1..n} 1/k^2.

Original entry on oeis.org

1, 2, 5, 7, 9, 7, 10, 20, 18, 14, 22, 19, 18, 24, 26, 24, 30, 30, 28, 37, 25, 30, 35, 35, 34, 38, 47, 52, 49, 54, 40, 49, 49, 69, 57, 67, 78, 67, 67, 68, 67, 64, 65, 86, 76, 81, 92, 79, 83, 83, 95, 82, 85, 80, 84, 95, 92, 91, 121, 105, 100, 108, 111, 109, 118, 105, 110, 88

Offset: 1

Benoit Cloitre, May 18 2002

Sum_{k>=1} 1/k^2 = zeta(2) = Pi^2/6.

			The simple continued fraction for Sum_{k=1..10} 1/k^2 is [1, 1, 1, 4, 1, 1, 10, 4, 1, 2, 5, 2, 1, 24] which contains 14 terms, hence a(10) = 14.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A013679, A055573, A007406, A007407.

lcf[f_] := Length[ContinuedFraction[f]]; lcf /@ Accumulate[Table[1/k^2, {k, 1, 100}]] (* Amiram Eldar, Apr 30 2022 *)

for(n=1,100,print1(length(contfrac(sum(i=1,n,1/i^2))),","))

Limit_{n ->infinity} a(n)/n = C =1.6....

A103716 Numerators of sum_{k=1..n} 1/k^10 =: Zeta(10,n).

Original entry on oeis.org

1, 1025, 60526249, 61978938025, 605263128567754849, 605263138567754849, 170971856382109814342232401, 175075181098169912564190119249, 10338014371627802833957102351534201, 413520574906423083987893722912609

Offset: 1

Wolfdieter Lang, Feb 15 2005

a(n) gives the partial sums, Zeta(10,n), of Euler's Zeta(10). 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 A103717 and for the rationals Zeta(10,n) see the W. Lang link under A103345.

For k=1..9 see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052, A103345/A103346, A103347/A103348, A103349/A103350, A103351/A103352.

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

a(n) = numerator(sum_{k=1..n} 1/k^10).

G.f. for rationals Zeta(10, n): polylogarithm(10, x)/(1-x).

cp's OEIS Frontend

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

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

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A373440 Denominator of sum of reciprocals of square divisors of n.

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A035166 Let d(m) = denominator of Sum_{k=1..m} 1/k^2 and consider f(m) = product of primes which appear to odd powers in d(m); sequence lists m such that f(m) is different from f(m-1).

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