A099828 -id:A099828 - cp's OEIS frontend

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

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

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

A103347 Numerators of Sum_{k=1..n} 1/k^7 = Zeta(7,n).

Original entry on oeis.org

1, 129, 282251, 36130315, 2822716691183, 940908897061, 774879868932307123, 99184670126682733619, 650750755630450535274259, 650750820166709327386387, 12681293156341501091194786541177, 12681293507322704937269896541177

Offset: 1

Wolfdieter Lang, Feb 15 2005

a(n) gives the partial sums, Zeta(7,n), of Euler's Zeta(7). 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 A103348 and for the rationals Zeta(7,n) see the W. Lang link under A103345.

Robert Israel, Table of n, a(n) for n = 1..336

For k=1..6 see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052, A103345/A103346.

f:= n -> numer(Psi(6,n+1)/720 + Zeta(7)):
map(f, [$1..20]); # Robert Israel, Mar 28 2018

s=0;lst={};Do[s+=n^1/n^8;AppendTo[lst,Numerator[s]],{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[ HarmonicNumber[n, 7] // Numerator, {n, 1, 12}] (* Jean-François Alcover, Dec 04 2013 *)

a(n) = numerator(sum_{k=1..n} 1/k^7).

G.f. for rationals Zeta(7, n): polylogarithm(7, x)/(1-x).

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

A103349 Numerators of sum_{k=1..n} 1/k^8 = Zeta(8,n).

Original entry on oeis.org

1, 257, 1686433, 431733409, 168646292872321, 168646392872321, 972213062238348973121, 248886558707571775009601, 1632944749460578249437992161, 1632944765723715465050248417

Offset: 1

Wolfdieter Lang, Feb 15 2005

a(n) gives the partial sums, Zeta(8,n) of Euler's Zeta(8). 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 A103350 and for the rationals Zeta(8,n) see the W. Lang link under A103345.

For k=1..7 see: A001008/A002805, A007406/A007407, A007408/A007409, A007410/A007480, A099828/A069052, A103345/A103346, A103347/A103348.

s=0;lst={};Do[s+=n^1/n^9;AppendTo[lst,Numerator[s]],{n,3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 24 2009 *)
Table[ HarmonicNumber[n, 8] // Numerator, {n, 1, 10}] (* Jean-François Alcover, Dec 04 2013 *)
Accumulate[1/Range[10]^8]//Numerator (* Harvey P. Dale, Aug 11 2024 *)

a(n)=numerator(sum_{k=1..n} 1/k^8).

G.f. for rationals Zeta(8, n): polylogarithm(8, x)/(1-x).

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).

A120290 Numerator of generalized harmonic number H(p-1,2p)= Sum[ 1/k^(2p), {k,1,p-1}] divided by p^2 for prime p>3.

Original entry on oeis.org

2479157521, 159936660724017234488561, 1119583852472161859174156302552583713828739479026834819554843860744244189

Offset: 3

Alexander Adamchuk, Jul 08 2006

Generalized harmonic number is H(n,m)= Sum[ 1/k^m, {k,1,n} ]. The numerator of generalized harmonic number H(p-1,2p) is divisible by p^2 for prime p>3.

			With prime(3) = 5, a(3) = numerator[ 1 + 1/2^10 + 1/3^10 + 1/4^10 ] / 5^2 = 61978938025 / 25 = 2479157521.

Alexander Adamchuk, First 5 terms.
Eric Weisstein's World of Mathematics, Harmonic Number.
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem.

Cf. A119722, A099828, A099827, A001008, A007406, A007408, A007410.

Table[Numerator[Sum[1/k^(2*Prime[n]),{k,1,Prime[n]-1}]],{n,3,7}]/Table[Prime[n]^2,{n,3,7}]

a(n) = numerator[ Sum[ 1/k^(2*Prime[n]), {k,1,Prime[n]-1} ]] / Prime[n]^2 for n>2.

cp's OEIS Frontend

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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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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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.

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A103347 Numerators of Sum_{k=1..n} 1/k^7 = Zeta(7,n).

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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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A103349 Numerators of sum_{k=1..n} 1/k^8 = Zeta(8,n).

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