A058313 -id:A058313 - cp's OEIS frontend

A136683 Numbers k such that A136675(k) is prime.

2, 3, 4, 5, 6, 9, 20, 21, 29, 119, 132, 151, 351, 434, 457, 462, 572, 611, 930, 1107, 1157, 1452, 1515, 2838, 3997, 5346, 6463, 6725, 7664, 10234, 14168, 14299

Offset: 1

Alexander Adamchuk, Jan 16 2008

A136675(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j^3.

Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A058313, A119682, A120296, A136675, A136676, A136677.
Cf. A001008, A007406, A007408, A007410, A099828, A103345.
Cf. A136681, A136682, A136684, A136685, A136686.

Do[ f=Numerator[ Sum[ (-1)^(k+1)*1/k^3, {k,1,n} ] ]; If[ PrimeQ[f], Print[ {n,f} ] ], {n,1,151} ]
Flatten[Position[Numerator[Accumulate[Table[(-1)^(k+1) 1/k^3,{k,3000}]]],?PrimeQ] ] (* _Harvey P. Dale, Feb 12 2013 *)

isok(n) = ispseudoprime(numerator(sum(k=1, n, (-1)^(k+1) / k^3))); \\ Daniel Suteu, Mar 15 2019

More terms from Harvey P. Dale, Feb 12 2013
a(25)-a(28) from Amiram Eldar, Mar 15 2019
a(29)-a(32) from Robert Price, Apr 22 2019

A136684 Numbers k such that A120296(k) is prime.

Original entry on oeis.org

3, 5, 8, 11, 20, 38, 61, 65, 71, 80, 83, 93, 233, 704, 1649, 2909, 3417, 3634, 9371

Offset: 1

Alexander Adamchuk, Jan 16 2008

A120296(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j^4.

Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A058313, A119682, A136675, A120296, A136676, A136677.
Cf. A001008, A007406, A007408, A007410, A099828, A103345.
Cf. A136681, A136682, A136683, A136685, A136686.

Do[ f=Numerator[ Sum[ (-1)^(k+1)*1/k^4, {k,1,n} ] ]; If[ PrimeQ[f], Print[ {n,f} ] ], {n,1,100} ]
Select[Range[1000],PrimeQ[Numerator[Sum[(-1)^(k+1) 1/k^4,{k,#}]]]&] (* Harvey P. Dale, Aug 28 2012 *)

More terms from Harvey P. Dale, Aug 28 2012

a(15)-a(19) from Robert Price, Apr 23 2019

A136685 Numbers k such that A136676(k) is prime.

Original entry on oeis.org

2, 19, 51, 78, 84, 130, 294, 910, 2223, 2642, 3261, 4312, 4973, 7846, 9439

Offset: 1

Alexander Adamchuk, Jan 16 2008

A136676(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j^5.

Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A058313, A119682, A136675, A120296, A136676, A136677.
Cf. A001008, A007406, A007408, A007410, A099828, A103345.
Cf. A136681, A136682, A136683, A136684, A136686.

Do[ f=Numerator[ Sum[ (-1)^(k+1)*1/k^5, {k,1,n} ] ]; If[ PrimeQ[f], Print[ {n,f} ] ], {n,1,130} ]

a(7)-a(8) from Amiram Eldar, Mar 14 2019

a(9)-a(15) from Robert Price, Apr 16 2019

A136686 Numbers k such that A136677(k) is prime.

Original entry on oeis.org

19, 47, 164, 235, 504, 1109, 1112, 5134, 9222, 12803

Offset: 1

Alexander Adamchuk, Jan 16 2008

A136677(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j^6.

Eric Weisstein's World of Mathematics, Harmonic Number.

Cf. A058313, A119682, A136675, A120296, A136676, A136677. Cf. A001008, A007406, A007408, A007410, A099828, A103345. Cf. A136681, A136682, A136683, A136684, A136685.

Do[ f=Numerator[ Sum[ (-1)^(k+1)*1/k^6, {k,1,n} ] ]; If[ PrimeQ[f], Print[ {n,f} ] ], {n,1,130} ]

a(4)-a(5) from Hiroaki Yamanouchi, Sep 22 2014
a(6) from Amiram Eldar, Mar 14 2019
a(7)-a(9) from Robert Price, Apr 20 2019
a(10) from Michael S. Branicky, Nov 16 2024

A075829 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))/(d(n)x + a(n)) (in lowest terms) and a(n), b(n), c(n), d(n) are positive integers.

Original entry on oeis.org

1, 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

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]
Difference between the denominator and the numerator of the (n-1)-th alternating harmonic number Sum_{k=1..n-1} (-1)^(k+1)*1/k = A058313(n-1)/A058312(n-1). - Alexander Adamchuk, Jul 22 2006
From Petros Hadjicostas, May 06 2020: (Start)
Inspired by Michael Somos's result below, we established the following formulas (valid for n >= 2). All the denominators in the first three formulas are equal to A334958(n).
b(n) = A024167(n)/gcd(A024167(n-1), A024167(n)).
c(n) = A024168(n)/gcd(A024168(n-1), A024168(n)).
d(n) = A024167(n-1)/gcd(A024167(n-1), A024167(n)).
b(n) + c(n) = n*(d(n) + a(n)).
u(n) = (A024167(n)*x + A024168(n))/(A024167(n-1)*x + A024168(n-1)). (End)

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

Cf. A075827 (= b), A075828 (= c), A075830 (= d).

Cf. A024167, A024168, A058312, A058313, A334958.

Denominator[Table[Sum[(-1)^(k+1)*1/k,{k,1,n-1}],{n,1,30}]]-Numerator[Table[Sum[(-1)^(k+1)*1/k,{k,1,n-1}],{n,1,30}]] (* Alexander Adamchuk, Jul 22 2006 *)

u(n) = if(n<2, x, (n-1)^2/u(n-1)+1);
a(n) = polcoeff(denominator(u(n)), 0, x);

a(n) = A024168(n-1)/gcd(A024168(n-1), A024168(n)). - Michael Somos, Oct 29 2002
From Alexander Adamchuk, Jul 22 2006: (Start)
a(n) = A058312(n-1) - A058313(n-1) for n > 1 with a(1) = 1.
a(n) = denominator(Sum_{k=1..n-1} (-1)^(k+1)*1/k) - numerator(Sum_{k=1..n-1}(-1)^(k+1)*1/k). (End)

Name edited by Petros Hadjicostas, May 06 2020

A121594 Numbers k such that k does not divide the denominator of the k-th alternating Harmonic number.

Original entry on oeis.org

15, 28, 75, 77, 104, 187, 196, 203, 210, 222, 228, 235, 238, 328, 345, 375, 551, 620, 847, 888, 1036, 1107, 1204, 1349, 1352, 1372, 1391, 1430, 1457, 1469, 1470, 1498, 1666, 1687, 1855, 1875, 2133, 2301, 2425, 2440, 2556, 2678, 2948, 3179, 3337, 3477

Offset: 1

Alexander Adamchuk, Aug 09 2006

nonn

Indices k such that A119788(k) is not equal to 1.

Also indices k such that numerators of k*H'(k) = A119787(k) and H'(k) = A058313(k) are different (H'(k) is the alternating harmonic number H'(k) = Sum_{j=1..k} (-1)^(j+1)*1/j). The ratio of numerators A119787(k)/A058313(k) for k = 1..400 is given in A119788(k). A121595(k) = A119788(a(k)) is the compressed version of A119788(k) (all 1 entries are excluded).

Giovanni Resta, Table of n, a(n) for n = 1..900

Cf. A119788, A119787, A058313, A121595.

Cf. A058312 = Denominator of the n-th alternating harmonic number, Sum_{k=1..n} (-1)^(k+1)/k. A074791 = numbers k such that k does not divide the denominator of the k-th Harmonic number.

Do[H=Sum[(-1)^(i+1)*1/i, {i, 1, n}]; a=Numerator[n*H]; b=Numerator[H]; If[ !Equal[a,b],Print[{n,a/b}]],{n,1,6000}]
f=0;Do[f=f+(-1)^(n+1)/n;If[ !IntegerQ[Denominator[f]/n],Print[n]],{n,1,100}] (* Alexander Adamchuk, Jan 02 2007 *)

A128672 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 = 2.

Original entry on oeis.org

20, 42, 100, 110, 156, 272, 294, 342, 500, 506, 812, 930, 1210, 1332, 1640, 1806, 2028, 2058, 2162, 2500, 2756, 3422, 3660, 4422, 4624, 4970, 5256, 6162, 6498, 6806, 7832, 9312, 10100, 10506, 11026, 11342, 11638, 11772, 12500, 12656, 13310, 14406, 16002, 17030

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 all geometric progressions of the form (p-1)*p^k for k > 0 and some primes p > 3. Note the factorization of initial terms of {a(n)} = {4*5, 6*7, 4*5^2, 10*11, 12*13, 16*17, 6*7^2, 18*19, 4*5^3, 22*23, 28*29, 30*31, 10*11^2, 36*37, 40*41, 42*43, 12*13^2, 6*7^3, 46*47, 4*5^4, 52*53, 58*59, 60*61, 66*67, 16*17^2, 70*71, 72*73, 78*79, 18*19^2, 82*83, ...}. The smallest term that does not fit this pattern is 11026 = ((149-1)/2) * 149.

Eric Weisstein's World of Mathematics, Harmonic Number

Similar sequences for generalized harmonic numbers with different k: A125581 (k=1), A128673 (k=3), A128674 (k=4), A128675 (k=5); A128676 (k=6).
For the least numbers k > 0 such that k^n does not divide the denominator of H(k,n) nor the denominator of H'(k,n), see A128670. See also A128671(n) = A128670(prime(n)).
Cf. A001008, A002805, A058313, A058312, A007406, A007407, A119682, A125581, A126196, A126197, A128674, A128675, A128676, A128673, A128670, A129671.

k=2; 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,1,7000} ]

Edited and extended by Max Alekseyev, May 07 2010

A128673 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 = 3.

Original entry on oeis.org

94556602, 141834903, 189113204, 283669806, 450820422

Offset: 1

Alexander Adamchuk, Apr 18 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.

Note that {a(n)} contains the following geometric progressions: ((16843-1)/3)*16843^m found by Max Alekseyev, ((16843-1)/2)*16843^m found by Max Alekseyev, ((16843-1)*2/3)*16843^m, (16843-1)*16843^m, 20826*21647^m found by Max Alekseyev, ((2124679-1)/3)*2124679^m, ((2124679-1)/2)*2124679^m, ((2124679-1)*2/3)*2124679^m, (2124679-1)*2124679^m. Here {16843, 2124679} = A088164 are the only two currently known Wolstenholme Primes: primes p such that {2p-1} choose {p-1} == 1 mod p^4. See more details in Comments at A128672 and A125581.

Eric Weisstein's World of Mathematics, Harmonic Number
Eric Weisstein's World of Mathematics, Wolstenholme Prime

Cf. A001008, A002805, A058313, A058312.
Cf. A007406, A007407, A119682, A007410, A120296, A099828.
Cf. A125581, A126196, A126197, A128672, A128674, A128675, A128676, A128670, A128671.
Cf. A088164 (Wolstenholme primes).

k=3; 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, 450820422} ]

A128676 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 = 6.

Original entry on oeis.org

20, 100, 110, 156, 161, 272, 342, 345, 500, 506, 812, 930, 1210, 1332, 1640, 1806, 2028, 2162, 2500, 2756, 3051, 3422, 3660, 3703, 4422, 4624, 4970, 5256, 6162, 6498, 6806, 7832, 7935, 9312, 9605, 10100, 10506, 11342, 11638, 11772, 12500, 12656, 13310

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 all terms of geometric progressions of the form (p-1)*p^k, k > 0, for some primes p >= 5, such as 4*5^k, 7*23^k, 15*23^k, 27*113^k, etc. Note the factorization of initial terms of {a(n)} = {4*5, 4*5^2, 10*11, 12*13, 7*23, 16*17, 18*19, 15*23, 4*5^3, 22*23, 28*29, 30*31, 10*11^2, 36*37, 40*41, 42*43, 12*13^2, 46*47, 4*5^4, 52*53, 27*113, 58*59, 60*61, 7*23^2, ...}. See more details in Comments at A128672 and A125581.

Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A001008, A002805, A058313, A058312.

Cf. A007406, A007407, A119682, A007410, A120296, A099828, A125581, A126196, A126197, A128672, A128673, A128676.

k=6; 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,1,3703} ]

Edited and extended by Max Alekseyev, May 08 2010

A049281 Numerators of coefficients in power series for -log(1+x)*log(1-x).

Original entry on oeis.org

1, 5, 47, 319, 1879, 20417, 263111, 52279, 1768477, 33464927, 166770367, 3825136961, 19081066231, 57128792093, 236266661971, 7313175618421, 14606816124167, 102126365345729, 3774664307989373, 3771059091081773

Offset: 1

Benoit Cloitre, May 03 2002

			-log(1 + x)*log(1 - x) = x^2 + 5/12*x^4 + 47/180*x^6 + 319/1680*x^8 + ...

Muniru A Asiru, Table of n, a(n) for n = 1..300

Bisection of A058313. Cf. A069685 (denominators).

List(List([1..25],n->(1/n)*Sum([1..2*n-1],k->(-1)^(k-1)/k)),NumeratorRat); # Muniru A Asiru, Jun 01 2018

seq(numer(1/n*add( (-1)^k/(2*n-1-k),k = 0..2*n-2)), n = 1..20); # Peter Bala, Feb 21 2017

a(n)/A069685(n) = Integral_{x=0..1} x^(n-1)*log(1 + 1/sqrt(x)) dx = 1/n*Sum_{k=0..2*n-2} (-1)^k/(2*n-1-k).
From Peter Bala, Feb 21 2017: (Start)
a(n) = numerator((1/n)*Sum_{k=1..2*n-1} (-1)^(k-1)/k ). Cf. A058313.
a(n) = numerator((1/2)*binomial(2*n,n)*Sum_{k=0..n-1} (-1)^k* binomial(n-1,k)/(n + k)^2 ).
The coefficients in the expansion of log(1 + x)*log(1 - x) are given by (1/2)*binomial(2*n,n)*Integral_{x = 0..1} (x*(1 - x))^(n-1)*log(x) dx.
log(1 + x)*log(1 - x) = (1/2)*Integral_{z = 0..1} log(z)/(z*(1 - z)) * (1/sqrt( 1 - 4*x^2*z*(1 - z) ) - 1) dz. (End)

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A136683 Numbers k such that A136675(k) is prime.

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A136684 Numbers k such that A120296(k) is prime.

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A136685 Numbers k such that A136676(k) is prime.

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A136686 Numbers k such that A136677(k) is prime.

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A075829 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))/(d(n)*x + a(n)) (in lowest terms) and a(n), b(n), c(n), d(n) are positive integers.

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A121594 Numbers k such that k does not divide the denominator of the k-th alternating Harmonic number.

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A128672 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 = 2.

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A128673 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 = 3.

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A075829 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))/(d(n)x + a(n)) (in lowest terms) and a(n), b(n), c(n), d(n) are positive integers.