A058313 -id:A058313 - cp's OEIS frontend

A075830 Let u(1) = x and u(n+1) = (n^2/u(n)) + 1 for n >= 1; then a(n) is such that u(n) = (b(n)x + c(n))/(a(n)x + d(n)) (in lowest terms) and a(n), b(n), c(n), d(n) are positive integers.

0, 1, 1, 5, 7, 47, 37, 319, 533, 1879, 1627, 20417, 18107, 263111, 237371, 52279, 95549, 1768477, 1632341, 33464927, 155685007, 166770367, 156188887, 3825136961, 3602044091, 19081066231, 18051406831, 57128792093, 7751493599

Offset: 1

Benoit Cloitre, Oct 14 2002

nonn

For x real <> 1 - 1/log(2), Lim_{n -> infinity} abs(u(n)-n) = abs((x - 1)/(1 + (x - 1)*log(2))). [Corrected by Petros Hadjicostas, May 18 2020]
From Petros Hadjicostas, May 05 2020: (Start)
Given x > 0, u(n) = (A075827(n)*x + A075828(n))/(a(n)*x + A075829(n)) = (b(n)*x + c(n))/(a(n)*x + d(n)) with gcd(gcd(b(n), c(n)), gcd(a(n), d(n))) = 1 for each n >= 1.
Conjecture 1: Define the sequences (A(n): n >= 1) and (B(n): n >= 1) by A(n+1) = n^2/A(n) + 1 for n >= 2 with A(1) = infinity and A(2) = 1, and B(n+1) = n^2/B(n) + 1 for n >= 3 with B(1) = 0, B(2) = infinity, and B(3) = 1. Then a(n) = denominator(A(n)), b(n) = numerator(A(n)), c(n) = numerator(B(n)), and d(n) = denominator(B(n)) (assuming infinity = 1/0). Also, gcd(a(n), d(n)) = 1.
In 2002, Michael Somos claimed that d(n) = A024168(n-1)/gcd(A024168(n-1), A024168(n)) for n >= 2. In 2006, N. J. A. Sloane claimed that a(n) = A058313(n-1) for n >= 2 while Alexander Adamchuk claimed that d(n) = A058312(n-1) - A058313(n-1) for n >= 2.
Conjecture 2: a(n) = A024167(n-1)/gcd(A024167(n-1), A024167(n)).
Conjecture 3: b(p) = a(p+1) for p = 1 or prime. In general, it seems that b(n) = A048671(n)*a(n+1) for all n for which A048671(n) < n.
Conjecture 4: c(n) = n*(a(n) + d(n)) - b(n) for n >= 1. (End)
All conjectures are proved in the link below except for the second part of Conjecture 3. - Petros Hadjicostas, May 21 2020

Petros Hadjicostas, Proofs of various results about the sequence u(n), 2020.

Apart from the leading term, same as A058313.
Cf. A075827 (= b), A075828 (= c), A075829 (= d).
Cf. A024167, A024168, A048671, A058312.

u(n)=if(n<2,x,(n-1)^2/u(n-1)+1);
a(n)=polcoeff(denominator(u(n)),1,x);

Name edited by Petros Hadjicostas, May 04 2020

A334958 GCD of consecutive terms of the factorial times the alternating harmonic series.

Original entry on oeis.org

1, 1, 1, 2, 2, 12, 12, 48, 144, 1440, 1440, 17280, 17280, 241920, 18144000, 145152000, 145152000, 2612736000, 2612736000, 10450944000, 219469824000, 4828336128000, 4828336128000, 115880067072000, 579400335360000, 15064408719360000, 135579678474240000, 26573616980951040000, 26573616980951040000

Offset: 1

Petros Hadjicostas, May 17 2020

nonn

For n = 1..14, we have a(n) = A025527(n), but a(15) = 18144000 <> 3628800 = A025527(15).
It appears that A025527(n) | a(n) for all n >= 1 and A025527(n) = a(n) for infinitely many n. In addition, it seems that a(n)/a(n-1) = A048671(n) for infinitely many n >= 2. However, I have not established these claims.
This sequence appears in formulas for sequences A075827, A075828, A075829, and A075830 (the first one of which was established in 2002 by Michael Somos).
Conjecture: a(n) = n! * Product_{p <= n} p^min(0, v_p(H'(n))), where the product ranges over primes p, H'(n) = Sum_{k=1..n} (-1)^(k+1)/k, and v_p(r) is the p-adic valuation of rational r (checked for n < 1100).

			A024167(4) = 4!*(1 - 1/2 + 1/3 - 1/4) = 14, A024167(5) = 5!*(1 - 1/2 + 1/3 - 1/4 + 1/5) = 94, A024168(4) = 4!*(1/2 - 1/3 + 1/4) = 10, and A024168(5) = 5!*(1/2 - 1/3 + 1/4 - 1/5) = 26. Then a(4) = gcd(14, 94) = gcd(10, 26) = gcd(14, 4!) = gcd(10, 4!) = gcd(14, 10) = 2.

Cf. A000142, A024167, A024168, A025527, A048671, A058312, A058313, A075827, A075828, A075829, A075830.

Cf. A056612 (similar sequence for the harmonic series).

b:= proc(n) b(n):= (-(-1)^n/n +`if`(n=1, 0, b(n-1))) end:
a:= n-> (f-> igcd(b(n)*f, f))(n!):
seq(a(n), n=1..30);  # Alois P. Heinz, May 18 2020

b[n_] := b[n] = -(-1)^n/n + If[n == 1, 0, b[n-1]];
a[n_] := GCD[b[n] #, #]&[n!];
Array[a, 30] (* Jean-François Alcover, Oct 27 2020, after Alois P. Heinz *)

def A():
    a, b, n = 1, 1, 2
    while True:
        yield gcd(a, b)
        b, a = a, a + b * n * n
        n += 1
a = A(); print([next(a) for  in range(29)]) # _Peter Luschny, May 19 2020

a(n) = gcd(A024167(n+1), A024167(n)) = gcd(A024168(n+1), A024168(n)) = gcd(A024167(n), n!) = gcd(A024168(n), n!) = gcd(A024167(n), A024168(n)).

A117664 Denominator of the sum of all elements in the n X n Hilbert matrix M(i,j) = 1/(i+j-1), where i,j = 1..n.

Original entry on oeis.org

1, 3, 10, 105, 252, 2310, 25740, 9009, 136136, 11639628, 10581480, 223092870, 1029659400, 2868336900, 11090902680, 644658718275, 606737617200, 4011209802600, 140603459396400, 133573286426580, 5215718803323600

Offset: 1

Alexander Adamchuk, Apr 11 2006

nonn

Sum_{j=1..n} Sum_{i=1..n} 1/(i+j-1) = A117731(n) / A117664(n) = 2n * H'(2n) = 2n * A058313(2n) / A058312(2n), where H'(2n) is 2n-th alternating sign Harmonic Number. H'(2n) = H(2n) - H(n), where H(n) is n-th Harmonic Number. - Alexander Adamchuk, Apr 23 2006

			For n=2, the 2 X 2 Hilbert matrix is [1, 1/2; 1/2, 1/3], so a(2) = denominator(1 + 1/2 + 1/2 + 1/3) = denominator(7/3) = 3.
The n X n Hilbert matrix begins:
    1 1/2 1/3 1/4  1/5  1/6  1/7  1/8 ...
  1/2 1/3 1/4 1/5  1/6  1/7  1/8  1/9 ...
  1/3 1/4 1/5 1/6  1/7  1/8  1/9 1/10 ...
  1/4 1/5 1/6 1/7  1/8  1/9 1/10 1/11 ...
  1/5 1/6 1/7 1/8  1/9 1/10 1/11 1/12 ...
  1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 ...
  ...

Eric Weisstein's World of Mathematics, Hilbert Matrix.
Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A091342, A098118, A111876, A082687, A086881, A005249, A001008, A002805.

Numerator is A117731(n).

Table[Denominator[Sum[1/(i + j - 1), {i, n}, {j, n}]], {n, 30}]

a(n) = A111876(n-1)/n.
a(n) = denominator( Sum_{j=1..n} Sum_{i=1..n} 1/(i+j-1) ). Numerator is A117731(n). - Alexander Adamchuk, Apr 23 2006
a(n) = denominator( Sum_{k=1..n} (2*k)/(n+k) ). - Peter Bala, Oct 10 2021

A119787 Numerator of the product of n and the n-th alternating harmonic number, Sum_{k=1..n} (-1)^(k+1)/k.

Original entry on oeis.org

1, 1, 5, 7, 47, 37, 319, 533, 1879, 1627, 20417, 18107, 263111, 237371, 261395, 95549, 1768477, 1632341, 33464927, 155685007, 166770367, 156188887, 3825136961, 3602044091, 19081066231, 18051406831, 57128792093, 54260455193

Offset: 1

Alexander Adamchuk, Jun 26 2006

a(n) almost always equals A058313(n), which is the numerator of the n-th alternating harmonic number, Sum ((-1)^(k+1)/k, k=1..n), except for n = 15, 28, 75, 77, 104, ... The ratio a(n)/A058313(n) for n = 1..400 is given in A119788.

			The first few fractions are 1, 1, 5/2, 7/3, 47/12, 37/10, 319/60, 533/105, 1879/280, ... = A119787/A334721. - _Petros Hadjicostas_, May 08 2020

Cf. A058313, A119788, A334721 (denominators).

Numerator[Table[Sum[(-1)^(i+1)*n/i, {i, 1, n}],{n,1,50}]]

a(n) = numerator(n*sum(k=1, n, (-1)^(k+1)/k)); \\ Michel Marcus, May 09 2020

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)*n/k).

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

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

A119788 Ratio of the numerator of the product of n and the n-th alternating harmonic number nH'(n) to the numerator of the n-th alternating harmonic number H'(n) = Sum_{k=1..n} (-1)^(k+1)1/k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13

Offset: 1

Alexander Adamchuk, Jun 26 2006, Sep 21 2006

Indices n such that a(n) is not equal to 1 are listed in A121594.
It appears that most a(n) > 1 are a prime divisor of their corresponding indices A121594(n). The first and only composite term up to a(6000) is a(1470) = 49 that also divides its index.
A compressed version of this sequence (all 1 entries are excluded) is A121595.

Alexander Adamchuk, Table of n, a(n) for n = 1..400

Cf. A058313, A092579, A119787, A121594, A121595.

Numerator[Table[n*Sum[(-1)^(i+1)*1/i, {i, 1, n}],{n,1,600}]]/Numerator[Table[Sum[(-1)^(i+1)*1/i, {i, 1, n}], {n,1,600}]]

a(n) = numerator(n*Sum_{i=1..n} (-1)^(i+1)*1/i) / numerator(Sum_{i=1..n}(-1)^(i+1)*1/i).

a(n) = A119787(n) / A058313(n).

A120301 Absolute value of numerator of the sum of all elements of the n X n matrix M with M[i,j] = (-1)^(i+j)*i/j for i,j = 1..n.

Original entry on oeis.org

1, 1, 5, 7, 47, 37, 319, 533, 1879, 1627, 20417, 18107, 263111, 237371, 52279, 95549, 1768477, 1632341, 167324635, 155685007, 166770367, 156188887, 3825136961, 3602044091, 19081066231, 18051406831, 57128792093, 54260455193

Offset: 1

Alexander Adamchuk, Jul 12 2006

Up to n = 18, a(n) is the same as A058313(n) = numerator of the n-th alternating harmonic number, Sum_{k=1..n} (-1)^(k+1)/k. a(n) differs from A058313(n) only for n = 18, 28, 87, 99.
Up to n = 100 the ratio a(n)/A058313(n) = {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1}.
A Wolstenholme-like theorem: for prime p > 3, if p = 6*k - 1, then p divides a(4*k-1), otherwise if p = 6*k + 1, then p divides a(4*k).
Sum_{j=1..n} Sum_{i=1..n} (-1)^(i+j)*i/j = -1/4 * (2*(-1)^n*n + (-1)^n - 1) * Sum_{k=1..n} (-1)^(k+1)/k.

			The absolute values of the first few fractions are 1, 1/2, 5/3, 7/6, 47/20, 37/20, 319/105, 533/210, 1879/504, ... = A120301/A334724. - _Petros Hadjicostas_, May 09 2020

Cf. A058313, A334724 (denominators)

Abs[Numerator[Table[Sum[Sum[(-1)^(i+j)*i/j,{i,1,n}],{j,1,n}],{n,1,50}]]]

a(n) = abs(numerator(sum(j=1, n, sum(i=1, n, (-1)^(i+j)*i/j)))); \\ Michel Marcus, May 09 2020

a(n) = abs(numerator(Sum_{j=1..n} Sum_{i=1..n} (-1)^(i+j)*i/j)).

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

cp's OEIS Frontend

A075830 Let u(1) = x and u(n+1) = (n^2/u(n)) + 1 for n >= 1; then a(n) is such that u(n) = (b(n)*x + c(n))/(a(n)*x + d(n)) (in lowest terms) and a(n), b(n), c(n), d(n) are positive integers.

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A334958 GCD of consecutive terms of the factorial times the alternating harmonic series.

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A117664 Denominator of the sum of all elements in the n X n Hilbert matrix M(i,j) = 1/(i+j-1), where i,j = 1..n.

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A119787 Numerator of the product of n and the n-th alternating harmonic number, Sum_{k=1..n} (-1)^(k+1)/k.

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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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Haskell

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A119788 Ratio of the numerator of the product of n and the n-th alternating harmonic number n*H'(n) to the numerator of the n-th alternating harmonic number H'(n) = Sum_{k=1..n} (-1)^(k+1)*1/k.

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A120301 Absolute value of numerator of the sum of all elements of the n X n matrix M with M[i,j] = (-1)^(i+j)*i/j for i,j = 1..n.

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A075830 Let u(1) = x and u(n+1) = (n^2/u(n)) + 1 for n >= 1; then a(n) is such that u(n) = (b(n)x + c(n))/(a(n)x + d(n)) (in lowest terms) and a(n), b(n), c(n), d(n) are positive integers.

A119788 Ratio of the numerator of the product of n and the n-th alternating harmonic number nH'(n) to the numerator of the n-th alternating harmonic number H'(n) = Sum_{k=1..n} (-1)^(k+1)1/k.