A058312 -id:A058312 - cp's OEIS frontend

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.

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

A025530 a(n) = (1/1 - 1/2 + ... + (-1)^(n-1)/n)*lcm{1..n}.

Original entry on oeis.org

1, 1, 5, 7, 47, 37, 319, 533, 1879, 1627, 20417, 18107, 263111, 237371, 261395, 477745, 8842385, 8161705, 167324635, 155685007, 166770367, 156188887, 3825136961, 3602044091, 19081066231, 18051406831, 57128792093, 54260455193, 1653866633797

Offset: 1

Clark Kimberling

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for sequences related to lcm's

Cf. A058312, A058313, A003418, A033999.

a025530 n = sum $ map (div (a003418 $ fromInteger n))
                      (zipWith (*) [1..n] a033999_list)
-- Reinhard Zumkeller, Dec 23 2011

nn=30;With[{fr=Accumulate[Table[1/(n (-1)^(n-1)),{n,nn}]]}, Table[fr[[n]] LCM@@ Range[n],{n,nn}]] (* Harvey P. Dale, Dec 27 2012 *)

a(n)=my(v=primes(primepi(n)),k=sqrtint(n),L=log(n+.5),t);t=prod(i=1,#v,if(v[i]>k,v[i],v[i]^(L\log(v[i]))));-sum(i=1,n,(-1)^i*t/i) \\ Charles R Greathouse IV, Dec 23 2011

s=1;v=vector(10^4,i,1);for(n=2,#v,t=n/gcd(s,n);s*=t;v[n]=v[n-1]*t-(-1)^n*s/n);v \\ Charles R Greathouse IV, Dec 23 2011

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

A130679 a(n) = (n+1+(-1)^n)*A024167(n), related to alternating harmonic sums.

Original entry on oeis.org

1, 4, 15, 84, 470, 3552, 26796, 255840, 2435184, 28114560, 323405280, 4380445440, 59105255040, 918796677120, 14228252640000, 249644312064000, 4363865549568000, 85297521899520000, 1661265370695168000

Offset: 1

Paul Curtz, Jun 29 2007

nonn

Inspired by a formula in the reference, the study of the singular points of planar differential systems leads to 3 two-dimensional polynomial families, one ordinary (degenerate case, considered in one dimension, see A129326) and two odd (the second, considered in one dimension, see A129587).
The first is in one dimension P(2n-1,x)=(n+1+x^n)*sum_{i=0..n-1} x^i/(i+1), n>=1.
The table of coefficients of P() with 2n coefficients per row starts:
2, 1;
3, 3/2, 1, 1/2;
4, 2, 4/3, 1, 1/2, 1/3;.. .
Rows multiplied by n!, the table becomes Q():
2, 1;
6, 3, 2, 1;
24, 12, 8, 6, 3, 2;
120, 60, 40, 30, 24, 12, 8, 6;
720, 360, 240, 180, 144,...
The sequence gives the alternating row sums of this table Q, positive sign for coefficients in front of even and negative sign for coefficients in front of odd powers of x.
The row sums of Q are (n+2)*A000254(n)= 3, 12, 55, 300...
Adding the alternating and ordinary row sums yields the sequence 4, 16, 70, 384....
The sequence of sums of antidiagonals in the Q table starts 2, 6+1=7, 24+3=27, 120+12+1=134.

			a(1) = 2-1.
a(2) = 6-3+2-1.
a(3) = 24-12+8-6+3-2.

G. C. Greubel, Table of n, a(n) for n = 1..250
Jean-François Alcover, Roots of P(2n-1,x) lie close to a circle (example)
P. Curtz, Stabilité locale des systèmes quadratiques, Ann. Sc. Ec. Norm. Sup., vol. 13 no. 3 (1980) 293-302.

a[n_] := (1/2)*(n + (-1)^n + 1)*n!*((-1)^n*(HarmonicNumber[(n-1)/2] - HarmonicNumber[n/2]) + Log[4]); Table[a[n] // FullSimplify, {n, 1, 19}] (* Jean-François Alcover, Oct 03 2012 *)

a(n) = n!*(n+1+(-1)^n)*A058313(n)/A058312(n). - R. J. Mathar, Jul 28 2008

Edited and extended by R. J. Mathar, Jul 28 2008

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

A262031 Numerator of partial sums of a reordered alternating harmonic series.

Original entry on oeis.org

1, 4, 5, 31, 247, 389, 1307, 15637, 13327, 187111, 199123, 353201, 6364777, 127056883, 23083451, 24191987, 579694957, 535076383, 13912332463, 43224283189, 40355946289, 1210479158981, 38689398709811, 72866186391697, 75054119011297, 77117026909777, 73105817107177, 2777117009412349

Offset: 0

Wolfdieter Lang, Sep 08 2015

For the denominators see A262022.
The reordered alternating harmonic series considered here is 1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + 1/9 + 1/11 - 1/6 + ... + ... - ...
The limit n -> infinity of the partial sums s(n) = a(n)/A262031(n) is 3*log(2)/2, approximately 1.03972077083991... For the decimal expansion see A262023.
Combining three consecutive terms of this series leads to the series b(0) + b(1) + ..., with b(k) = (1/2)*(8*k+5)/((4*k+1)*(4*k+3)*(k+1)). This produces partial sums 5/6, 13/140, 7/198, 29/1560, 37/3230, ..., which are given by s(3*n+2), n = 0, 1, .... Therefore, the limit is the same as the one given above, and it is obtained from Sum_{k=0..n} b(k) = (1/4)*Psi(n+5/4) + (1/4)*Psi(n+7/4) - (1/2)*Psi(n+2) + (3/2)*log(2), with the digamma function Psi(x).
This reordered alternating harmonic series appears as an example in the famous Dirichlet article, p. 319 (Werke I). Martin Ohm showed that for the reordering with alternating m consecutive positive terms followed by n negative terms (here n = 2 and m = 1) the sum becomes log(2) + (1/2)*log(m/n). See the reference, paragraph 8. p. 12-14. See also the Pringsheim reference.

			The first fractions s(n) (in lowest terms) are 1, 4/3, 5/6, 31/30, 247/210, 389/420, 1307/1260, 15637/13860, 13327/13860, 187111/180180, 199123/180180, 353201/360360, ...
The values s(10^n), for n=0..6, are (Maple 10 digits) [1.333333333, 1.105133755, 1.047114258, 1.040469694, 1.039795760, 1.039728271, 1.039721521], to be compared with 3*log(2)/2 (approximately 1.039720771).

P. G. Lejeune Dirichlet, Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Faktor sind, unendlich viele Primzahlen enthält, Abh. Preuss. Akad. Wiss. (1837) 45 -81; Werke I, 313-342.
M. Ohm, De nonnullis seriebus infinitis summandis, Berolini, 1839, Typis Trowitzschii et filli.
A. Pringsheim, Ueber die Werthveränderungen bedingt convergenter Reihen und Producte, Math. Ann. 22 (1838) 455-503.

Cf. A262022 (denominator), A262023, A058313, A058312, A002162.

Table[Numerator@ Sum[Which[Mod[k, 3] == 0, 3/(4 k + 3), Mod[k, 3] == 1, 3/(4 k + 5), True, -3/(2 (k + 1))], {k, 0, n} ], {n, 0, 27}] (* Michael De Vlieger, Jul 26 2016 *)

lista(nn) = {my(s = 0); for (k=0, nn, if (k%3==2, t = -3/(2*(k+1)), if (k%3==1, t = 3/(4*k+5), t = 3/(4*k+3))); s += t; print1(numerator(s), ", "););} \\ Michel Marcus, Sep 13 2015

a(n) = numerator(s(n)) with s(n) = Sum_{k=0..n} c(k), where c(k) = 3/(4*k+3), 3/(4*k+5), -3/(2*(k+1)) if k == 0, 1, 2 (mod 3), respectively.

A305307 Expansion of e.g.f. 1/(1 - log(1 + x)/(1 - x)).

Original entry on oeis.org

1, 1, 3, 17, 120, 1084, 11642, 146446, 2101656, 33958344, 609431232, 12033015840, 259163792016, 6047213451408, 151953760489008, 4091057804809104, 117485988199385088, 3584814699783432960, 115816462543697120640, 3949619921174717629056, 141780511159572486530304, 5344008726418981985707776

Offset: 0

Ilya Gutkovskiy, May 29 2018

nonn

a(n)/n! is the invert transform of [1, 1 - 1/2, 1 - 1/2 + 1/3, 1 - 1/2 + 1/3 - 1/4, 1 - 1/2 + 1/3 - 1/4 + 1/5, ...].

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 120*x^4/4! + 1084*x^5/5! + 11642*x^6/6! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..410
N. J. A. Sloane, Transforms

Cf. A006252, A024167, A058312, A058313, A305306.

g:= proc(n) g(n):= `if`(n=1, 0, g(n-1))-(-1)^n/n end:
b:= proc(n) option remember; `if`(n=0, 1,
      add(g(j)*b(n-j), j=1..n))
    end:
a:= n-> b(n)*n!:
seq(a(n), n=0..20);  # Alois P. Heinz, May 29 2018

nmax = 21; CoefficientList[Series[1/(1 - Log[1 + x]/(1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[1/(1 - Sum[Sum[(-1)^(j + 1)/j, {j, 1, k}] x^k , {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[((-1)^(k + 1) LerchPhi[-1, 1, k + 1] + Log[2]) a[n - k], {k, 1, n}]; Table[n! a[n], {n, 0, 21}]

E.g.f.: 1/(1 - Sum_{k>=1} (A058313(k)/A058312(k))*x^k).

a(n) ~ n! * (2 - LambertW(exp(2))) / ((1 + 1/LambertW(exp(2))) * (LambertW(exp(2)) - 1)^(n+1)). - Vaclav Kotesovec, Aug 08 2021

A364317 Irregular triangle T read by rows: T(n, k) gives the number of permutations of [n] = {1, 2, ..., n} with a cycle of length m = floor(n/2) + k = A138099(n, k), for 1 <= k <= n - floor(n/2) = ceiling(n/2).

Original entry on oeis.org

1, 1, 3, 2, 8, 6, 40, 30, 24, 180, 144, 120, 1260, 1008, 840, 720, 8064, 6720, 5760, 5040, 72576, 60480, 51840, 45360, 40320, 604800, 518400, 453600, 403200, 362880, 6652800, 5702400, 4989600, 4435200, 3991680, 3628800

Offset: 1

Wolfdieter Lang, Aug 12 2023

The length of row n is ceiling(n/2) = A008619(n-1).
The numbers for these cycles of permutations of [n], appear in the solution of the Locker Problem. See the link, p. 25.
For the probability of failures with the strategy used in the locker problem with n lockers and opening of up to floor(n/2) lockers see A058313(n)/A058312(n), for n > = 1. For n = 1 the one team member is not allowed to open the one locker (with the member's wallet) because (n/2) = 0; so certainly a failure.
For the probability of success in this locker problem for n lockers see A119248(n)/A058312(n), for n >= 1.

			The irregular triangle begins:
n\k       1       2       3       4       5       6 ...
-------------------------------------------------------
1:        1
2:        1
3:        3       2
4:        8       6
5:       40      30      24
6:      180     144     120
7:     1260    1008     840     720
8:     8064    6720    5760    5040
9:    72576   60480   51840   45360   40320
10:  604800  518400  453600  403200  362880
11: 6652800 5702400 4989600 4435200 3991680 3628800
...
T(5, 1) = 40 because m(5, 1) = 2 + 1 = 3, and for each of the binomial(5, 3) = 10 possibilities for choosing three numbers from [5] there are (3 - 1)! = 2 3-cycles if each starts with the smallest number, e.g., for {2, 3, 5} the cycles are (2, 3, 5) and (2, 5, 3). For the remaining 5-3 = 2 numbers there are 2! possible permutations; in the example permutations of {1, 4}, namely (1)(4) and (1,4). Thus T(5, 3) = binomial(5, 3)*2!*2! = 10*2*2 = 40 = 5!/3.

Paolo Xausa, Table of n, a(n) for n = 1..1024 (rows 1..63 of the triangle, flattened)
Eugene Curtis and Max Warshauer, The Locker Puzzle, The Mathematical Intelligencer 28 (2006), pp. 28-31.
Christoph Pöppe, Freiheit für die Kombinatoriker, Spektrum, Highlights 1.21, pp. 16-18 (in German).
Wikipedia, 100 prisoners problem.
Wikipedia (in German), Problem der 100 Gefangenen.

Cf. A008619, A058313/A058312, A119248/A058312, A126074, A138099.

A364317row[n_]:=n!/(Floor[n/2]+Range[Ceiling[n/2]]);Array[A364317row,10] (* Paolo Xausa, Sep 25 2023 *)

T(n, k) = binomial(n, m(n, k))*(m(n, k) - 1)!*(n - m(n, k))! = n!/m(n, k), with m(n, k) = floor(n/2) + k = A138099(n, k), for n >= 1 and k = 1, 2, ..., ceiling(n/2).

A082688 Denominator of Sum_{k=1..n} 1/(n+k).

Original entry on oeis.org

2, 12, 60, 840, 2520, 27720, 360360, 144144, 2450448, 232792560, 232792560, 5354228880, 26771144400, 11473347600, 332727080400, 20629078984800, 20629078984800, 144403552893600, 5342931457063200, 5342931457063200

Offset: 1

Benoit Cloitre, Apr 12 2003

			1/2, 7/12, 37/60, 533/840, 1627/2520, 18107/27720, 237371/360360, ...

T. D. Noe, Table of n, a(n) for n=1..100

Cf. A058312, A082687 (numerators).

Table[HarmonicNumber[2 n] - HarmonicNumber[n], {n, 20}] // Denominator (* Eric W. Weisstein, Dec 14 2017 *)

a(n) = denominator(sum(k=1, n, 1/(n+k))); \\ Michel Marcus, Dec 14 2017

limit n ->infinity Sum_{k=1..n} 1/(n+k) = log(2).

Denominator of Psi(2*n+1)-Psi(n+1). - Vladeta Jovovic, Aug 24 2003

cp's OEIS Frontend

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.

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A025530 a(n) = (1/1 - 1/2 + ... + (-1)^(n-1)/n)*lcm{1..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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A130679 a(n) = (n+1+(-1)^n)*A024167(n), related to alternating harmonic sums.

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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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A262031 Numerator of partial sums of a reordered alternating harmonic series.

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A305307 Expansion of e.g.f. 1/(1 - log(1 + x)/(1 - x)).

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