A058313 -id:A058313 - cp's OEIS frontend

A136681 Numbers k such that A058313(k) is prime.

3, 4, 5, 6, 9, 10, 13, 16, 17, 18, 37, 43, 58, 121, 124, 126, 137, 203, 247, 283, 285, 286, 289, 317, 424, 508, 751, 790, 937, 958, 1066, 1097, 1151, 1166, 1194, 1199, 1235, 1414, 1418, 1460, 1498, 1573, 2090, 2122, 2691, 2718, 3030, 3426, 3600, 3653, 3737

Offset: 1

Alexander Adamchuk, Jan 16 2008

nonn

A058313(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j.

Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A058313, A119682, A120296, A136675, A136676, A136677.
Cf. A001008, A007406, A007408, A007410, A099828, A103345.
Cf. A136682, A136683, A136684, A136685, A136686.

Do[ f=Numerator[ Sum[ (-1)^(k+1)*1/k, {k,1,n} ] ]; If[ PrimeQ[f], Print[ {n,f} ] ], {n,1,317} ]

isok(n) = isprime(numerator(sum(k=1, n, (-1)^(k+1)/k))); \\ Michel Marcus, Mar 14 2019

a(25)-a(30) from James R. Buddenhagen, Sep 22 2015

a(31)-a(51) from Amiram Eldar, Mar 14 2019

A119248 a(n) is the difference between denominator and numerator of the n-th alternating harmonic number Sum_{k=1..n} (-1)^(k+1)/k = A058313(n)/A058312(n).

Original entry on oeis.org

0, 1, 1, 5, 13, 23, 101, 307, 641, 893, 7303, 9613, 97249, 122989, 19793, 48595, 681971, 818107, 13093585, 77107553, 66022193, 76603673, 1529091919, 1752184789, 7690078169, 8719737569, 23184641107, 3721854001, 96460418429

Offset: 1

Alexander Adamchuk, Jul 22 2006

a(n)/A058312(n) = 1 - A058313(n)/A058312(n) appears in the locker puzzle (see the links in A364317) for the probability of success with the strategy used there for n lockers and allowed openings of up to floor(n/2) lockers. Note that gcd(a(n), A058312(n)) = 1. - Wolfdieter Lang, Aug 12 2023

Cf. A058312, A058313, A075829, A364317.

Denominator[Table[Sum[(-1)^(k+1)/k,{k,1,n}],{n,1,30}]]-Numerator[Table[Sum[(-1)^(k+1)/k,{k,1,n}],{n,1,30}]]

a(n) = my(x=sum(k=1, n, (-1)^(k+1)/k)); denominator(x) - numerator(x); \\ Michel Marcus, May 07 2020

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/k) - numerator(Sum_{k=1..n} (-1)^(k+1)/k).
a(n) = A058312(n) - A058313(n).
a(n) = A075829(n+1).
a(n) = numerator(Sum_{k=2..n} (-1)^k/k). (Cf. A024168.) - Petros Hadjicostas, May 17 2020

A123944 Numbers k such that A120301(k) differs from A058313(k).

Original entry on oeis.org

19, 28, 87, 99, 104, 196, 203, 210, 222, 228, 231, 238, 281, 328, 367, 499, 579, 620, 888, 967, 1036, 1147, 1204, 1352, 1372, 1403, 1419, 1430, 1470, 1481, 1498, 1503, 1666, 1693, 1907, 2211, 2359, 2440, 2499, 2521, 2556, 2678, 2948, 3407, 3467, 3504, 3537, 3892, 4046, 4079, 4108

Offset: 1

Alexander Adamchuk, Nov 22 2006

nonn

The ratio A120301(n)/A058313(n) = 1 for most n.
The ratio A120301(a(n))/A058313(a(n)) = {5, 7, 11, 5, 13, 7, 17, 7, 37, 10, 29, 119, 47, 41, 23, 5, 29, 31, 37, 11, 37, 41, 43, 13, 7, 13, 71, 13, 49, 13, 7,...} is prime for the most a(n).
The first composite ratio A120301(a(n))/A058313(a(n)) corresponds to a(n) = a(29) = 1470 because A120301(1470)/A058313(1470) = 49 = 7^2. [Edited by Petros Hadjicostas, May 09 2020]

Cf. A058313, A120301.

f=0; Do[f=f+(-1)^(n+1)*1/n; g=Abs[(2(-1)^n*n+(-1)^n-1)/4]*f; rfg=Numerator[g]/Numerator[f]; If[(rfg==1)==False, Print[{n,rfg}]], {n,1,15000}]

isok(n) = my(sn = sum(k=1, n, (-1)^(k+1)/k)); numerator(sn) != abs(numerator((-1/4) * (2*(-1)^n*n + (-1)^n - 1)*sn));
for (n=1, 4200, if (isok(n), print1(n, ", "))); \\ Michel Marcus, May 10 2020

a(47)-a(51) from Petros Hadjicostas, May 09 2020

A128465 Numbers k such that k divides the numerator of alternating Harmonic number H'((k+1)/2) = A058313((k+1)/2).

Original entry on oeis.org

1, 5, 7, 71, 379, 2659

Offset: 1

Alexander Adamchuk, Mar 10 2007

For k > 1 all 5 listed terms are primes.
The only known numbers k such that k divides the numerator of alternating Harmonic number H'((k-1)/2) = A058313((k-1)/2) are the Wieferich primes (A001220): 1093 and 3511.
An odd prime p = prime(n) belongs to this sequence iff Fermat quotient A007663(n) == A130912(n) == 2*(-1)^((p+1)/2) (mod p). - Max Alekseyev, Nov 30 2022

Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A001008, A007663, A001220, A058313, A125854, A121999, A130912.

f=0; Do[ f = f + (-1)^(n+1)*1/n; g = Numerator[f]; If[ IntegerQ[ g/(2n-1) ], Print[2n-1]], {n,1,3000} ]

Edited by Max Alekseyev, Nov 30 2022

A351789 Decimal expansion of Sum_{k>=1} AH(k)*F(k)/2^k, where AH(k) = A058313(k)/A058312(k) is the k-th alternating harmonic number and F(k) = A000045(k) is the k-th Fibonacci number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Feb 19 2022

			1.51437037420622187243459478916165077964831313316887...

Seán M. Stewart, Problem H-893, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 60, No. 1 (2022), p. 91.

Cf. A000045, A001622, A002390, A058312, A058313, A349850, A351794.

RealDigits[Log[5/4] + 6*Log[GoldenRatio]/Sqrt[5], 10, 100][[1]]

Equals log(5/4) + 6*log(phi)/sqrt(5), where phi is the golden ratio (A001622) (Stewart, 2022).

A351794 Decimal expansion of Sum_{k>=1} AH(k)*L(k)/2^k, where AH(k) = A058313(k)/A058312(k) is the k-th alternating harmonic number and L(k) = A000032(k) is the k-th Lucas number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Feb 19 2022

			2.82147535876264946174605143568253063724665666934546...

Cf. A000032, A001622, A002390, A058312, A058313, A349850, A349851, A351789.

RealDigits[3*Log[5/4] + 10*Log[GoldenRatio]/Sqrt[5], 10, 100][[1]]

Equals 3*log(5/4) + 10*log(phi)/sqrt(5), where phi is the golden ratio (A001622).

Equals (4/3)*log(5/4) + (5/3)*A351789.

A167372 a(n) = A120301(A123944(n))/A058313(A123944(n)).

Original entry on oeis.org

5, 7, 11, 5, 13, 7, 17, 7, 37, 19, 29, 119, 47, 41, 23, 5, 29, 31, 37, 11, 37, 41, 43, 13, 7, 13, 71, 13, 49, 13, 7, 47, 7, 7, 53, 79, 59, 61, 5, 97, 71, 103, 67, 71, 17, 73, 61, 139, 17, 17, 79, 19, 19, 19, 19, 83, 151, 89, 29, 97, 263, 29, 101, 103, 223, 107, 109, 271, 37, 23, 113, 359

Offset: 1

Alexander Adamchuk, Nov 02 2009

nonn

The ratio A120301(n)/A058313(n) = 1 for most n.
a(n) is prime for most n.
The first composite ratio a(12) = 119 = 7*17 corresponds to A123944(12) = 238.
The next two composite ratios a(29) = a(76) = 49 = 7^2 correspond to A123944(29) = 1470 and A123944(76) = 10290. [Edited by Petros Hadjicostas, May 09 2020]

Cf. A058313, A120301, A123944.

f = 0; Do[f = f + (-1)^(n + 1) * 1/n; g = Abs[(2(-1)^n * n + (-1)^n - 1)/4] * f; rfg = Numerator[g]/Numerator[f]; If[(rfg == 1) == False, Print[rfg]], {n, 1500}]

lista(nn) = {for (n=1, nn, my(sn = sum(k=1, n, (-1)^(k+1)/k)); if ((s=numerator(sn)) != (ss=abs(numerator((-1/4) * (2*(-1)^n*n + (-1)^n - 1) * sn))), print1(ss/s, ", ")););} \\ Michel Marcus, May 10 2020

a(32)-a(46) from Petros Hadjicostas, May 09 2020, using Michel Marcus's program and the data from A123944

a(47)-a(72) from Petros Hadjicostas, May 09 2020, using the Mathematica program

A173756 Partial sums of A058313.

Original entry on oeis.org

1, 2, 7, 14, 61, 98, 417, 950, 2829, 4456, 24873, 42980, 306091, 543462, 595741, 691290, 2459767, 4092108, 37557035, 193242042, 360012409, 516201296, 4341338257, 7943382348, 27024448579, 45075855410, 102204647503, 109956141102, 346222803073, 571398562364

Offset: 1

Jonathan Vos Post, Feb 23 2010

Partial sum of the numerator of the n-th alternating harmonic number.

Cf. A001008 (numerator of the n-th harmonic number), A025530, A058312 (denominators of the underlying sequence), A058313, A075830.

a := proc(n) local i, k:
add(numer(add((-1)^(k + 1)/k, k = 1 .. i)), i = 1 .. n): end proc:
seq(a(n), n = 1 .. 40); # Petros Hadjicostas, May 06 2020

a(n) = sum(i=1, n, numerator(sum(k=1, i, (-1)^(k+1)/k))); \\ Michel Marcus, May 07 2020

a(n) = Sum_{i=1..n} A058313(i) = Sum_{i=1..n} numerator(Sum_{k=1..i} (-1)^(k+1)/k). [Corrected by Petros Hadjicostas and Michel Marcus, May 06 2020]

Data corrected and extended by Petros Hadjicostas, May 06 2020

A058312 Denominator of the n-th alternating harmonic number, Sum_{k=1..n} (-1)^(k+1)/k.

Original entry on oeis.org

1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720, 360360, 360360, 72072, 144144, 2450448, 2450448, 46558512, 232792560, 232792560, 232792560, 5354228880, 5354228880, 26771144400, 26771144400, 80313433200, 11473347600

Offset: 1

N. J. A. Sloane, Dec 09 2000

a(n) is a divisor of A003418(n). The first time this is a proper divisor, is a(15); see A269626. - Jeppe Stig Nielsen, Mar 01 2016

			1, 1/2, 5/6, 7/12, 47/60, 37/60, 319/420, 533/840, 1879/2520, ...

T. D. Noe and Robert Israel, Table of n, a(n) for n = 1..2000 (1..200 from T. D. Noe)
Eric Weisstein's World of Mathematics, Harmonic Number

Numerators are A058313. Cf. A025530.
Cf. A002805 (denominator of n-th harmonic number).
Cf. A121594, A181983, A003418, A269626.

import Data.Ratio((%), denominator)
a058312 n = a058312_list !! (n-1)
a058312_list = map denominator $ scanl1 (+) $
                   map (1 %) $ tail a181983_list
-- Reinhard Zumkeller, Mar 20 2013

A058313 := n->denom(add((-1)^(k+1)/k,k=1..n));
# Alternatively:
a := n -> denom(harmonic(n) - harmonic((n-modp(n,2))/2)):
seq(a(n), n=1..28); # Peter Luschny, May 03 2016

a[n_] := Sum[(-1)^(k+1)/k, {k, 1, n}]; Table[a[n] // Denominator, {n, 1, 30}] (* Jean-François Alcover, May 26 2015 *)
a[n_]:= (-1)^n(HarmonicNumber[n/2-1/2]-HarmonicNumber[n/2]+(-1)^n Log[4])/2; Table[a[n] // FullSimplify, {n, 1, 29}] // Denominator (* Gerry Martens, Jul 05 2015 *)
Rest[Denominator[CoefficientList[Series[Log[1 + x]/(1 - x),{x, 0, 33}], x]]] (* Vincenzo Librandi, Jul 06 2015 *)

a(n)=denominator(polcoeff(-log(1-x)/(x+1)+O(x^(n+1)),n))

a(n)=denominator(sum(k=1,n,(-1)^(k+1)/k)) \\ Jeppe Stig Nielsen, Mar 01 2016

G.f. for A058313(n)/ A058312(n): log(1+x)/(1-x). - Benoit Cloitre, Jun 15 2003
a(n) = n*a(n-1)/gcd(n*a(n-1), n*A058313(n-1)+(-1)^(n-1)*a(n-1)). - Robert Israel, Jul 05 2015
a(n) = the (reduced) denominator of the continued fraction 1/(1 + 1^2/(1 + 2^2/(1 + 3^2/(1 + ... + (n-1)^2/(1))))). - Peter Bala, Feb 18 2024

A197070 Decimal expansion of the Dirichlet eta-function at 3.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Oct 09 2011

This constant is irrational by Apéry's theorem. - Charles R Greathouse IV, Feb 11 2024

			0.9015426773696957140498036211335874930737...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
R. Barbieri, J. A. Mignaco and E. Remiddi, Electron form factors up to fourth order. I., Il Nuovo Cim. 11A (4) (1972) 824-864 Table II (4)
Su Hu, Min-soo Kim, Euler's integral, multiple cosine function and zeta values, arXiv:2201.011247 (2023), series last equation.
Seán Stewart, Problem 12206, The American Mathematical Monthly, Vol. 127, No. 8 (2020), p. 752.
Wikipedia, Dirichlet eta function.

Cf. A002117 (zeta(3)), A058312, A058313, A072691, A136675, A233090 (5*zeta(3)/8), A233091 (7*zeta(3)/8), A334582.

Maple
```
3*Zeta(3)/4 ; evalf(%) ;
```

RealDigits[3(Zeta[3])/4, 10, 75][[1]] (* Bruno Berselli, Dec 20 2011 *)

-polylog(3,-1) \\ Charles R Greathouse IV, Mar 28 2012

3/4*zeta(3) \\ Charles R Greathouse IV, Mar 28 2012

Equals 3*zeta(3)/4 = 3*A002117/4.
Also equals the integral over the unit cube [0,1]x[0,1]x[0,1] of 1/(1+x*y*z) dx dy dz. - Jean-François Alcover, Nov 24 2014
Equals Sum_{n>=1} (-1)^(n+1)/n^3. - Terry D. Grant, Aug 03 2016
Equals Lim_{n -> infinity} A136675(n)/A334582(n). - Petros Hadjicostas, May 07 2020
Equals Sum_{n>=1} AH(2*n)/n^2, where AH(n) = Sum_{k=1..n} (-1)^(k+1)/k = A058313(n)/A058312(n) is the n-th alternating harmonic number (Stewart, 2020). - Amiram Eldar, Oct 04 2021
Equals -int_0^1 log(x)log(1+x)/x dx [Barbieri] - R. J. Mathar, Jun 07 2024

cp's OEIS Frontend

A136681 Numbers k such that A058313(k) is prime.

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A119248 a(n) is the difference between denominator and numerator of the n-th alternating harmonic number Sum_{k=1..n} (-1)^(k+1)/k = A058313(n)/A058312(n).

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A123944 Numbers k such that A120301(k) differs from A058313(k).

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A128465 Numbers k such that k divides the numerator of alternating Harmonic number H'((k+1)/2) = A058313((k+1)/2).

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A351789 Decimal expansion of Sum_{k>=1} AH(k)*F(k)/2^k, where AH(k) = A058313(k)/A058312(k) is the k-th alternating harmonic number and F(k) = A000045(k) is the k-th Fibonacci number.

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A351794 Decimal expansion of Sum_{k>=1} AH(k)*L(k)/2^k, where AH(k) = A058313(k)/A058312(k) is the k-th alternating harmonic number and L(k) = A000032(k) is the k-th Lucas number.

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A167372 a(n) = A120301(A123944(n))/A058313(A123944(n)).

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A173756 Partial sums of A058313.

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