A103345 -id:A103345 - cp's OEIS frontend

A136682 Numbers k such that A119682(k) is prime.

2, 3, 5, 8, 23, 41, 47, 48, 49, 95, 125, 203, 209, 284, 323, 395, 504, 553, 655, 781, 954, 1022, 1474, 1797, 1869, 1923, 1934, 1968, 2043, 2678, 3413, 3439, 4032, 4142, 4540, 4895, 5018, 5110, 5194, 5357, 6591, 11504, 11949, 14084, 20365

Offset: 1

Alexander Adamchuk, Jan 16 2008

nonn

A119682(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j^2.

Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A058313, A119682, A120296, A136675, A136676, A136677.
Cf. A001008, A007406, A007408, A007410, A099828, A103345.
Cf. A136681, A136683, A136684, A136685, A136686.

Do[ f=Numerator[ Sum[ (-1)^(k+1)*1/k^2, {k,1,n} ] ]; If[ PrimeQ[f], Print[ {n,f} ] ], {n,1,125} ]

a(12)-a(17) from Alexander Adamchuk, Apr 28 2008
a(18)-a(31) from Amiram Eldar, Mar 14 2019
a(32)-a(45) from Robert Price, Apr 14 2019

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

Original entry on oeis.org

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

A291456 a(n) = (n!)^6 * Sum_{i=1..n} 1/i^6.

Original entry on oeis.org

0, 1, 65, 47449, 194397760, 3037656102976, 141727869124448256, 16674281388691716870144, 4371079210518164503303028736, 2322975003299339366419974718488576, 2322977286679362958150790503464960000000

Offset: 0

Seiichi Manyama, Aug 24 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..104

Column k=6 of A291556.
Cf. A000254 (k=1), A001819 (k=2), A066989 (k=3), A203229 (k=4), A099827 (k=5).
Cf. A008516, A103345, A103346.

Table[(n!)^6 * Sum[1/i^6, {i, 1, n}], {n, 0, 15}] (* Vaclav Kotesovec, Aug 27 2017 *)

a(0) = 0, a(1) = 1, a(n+1) = (n^6 + (n+1)^6)*a(n) - n^12*a(n-1) for n > 0.
a(n) ~ 8 * Pi^9 * n^(6*n+3) / (945 * exp(6*n)). - Vaclav Kotesovec, Aug 27 2017
a(n) = (n!)^6 * A103345(n)/A103346(n). - Petros Hadjicostas, May 10 2020
Sum_{n>=0} a(n) * x^n / (n!)^6 = polylog(6,x) / (1 - x). - Ilya Gutkovskiy, Jul 15 2020

A120077 Denominators of row sums of rational triangle A120072/A120073.

Original entry on oeis.org

4, 36, 144, 3600, 3600, 176400, 705600, 6350400, 1270080, 153679680, 153679680, 25971865920, 25971865920, 129859329600, 519437318400, 150117385017600, 150117385017600, 54192375991353600, 2167695039654144, 1548353599752960, 221193371393280, 117011293467045120

Offset: 2

Wolfdieter Lang, Jul 20 2006

The first 19 terms coincide with A007407(n), for n>=2. However a(20) = 2167695039654144 and A007407(20) = 10838475198270720 = 5*a(20). Also a(21) = 1548353599752960 and A007407(21) = 221193371393280 = a(21)/7. From n = 22 up to at least n = 100 (checked) both sequences coincide again.
See the W. Lang link under A120072 for more details.
The corresponding numerators are given by A120076.
The n for which a(n) differs from A007407(n) are given by A309829. - Jeppe Stig Nielsen, Aug 18 2019

			The rationals A120076(m)/a(m), m>=2, begin with (3/4, 37/36, 169/144, 4549/3600, 4769/3600, ... ).

Jeppe Stig Nielsen, Table of n, a(n) for n = 2..1150

Cf. A007407, A120070, A120072, A120073, A120074, A120075, A120076, A309829.

A120077:= func< n | Denominator( (&+[1/k^2: k in [1..n]]) -1/n) >;
[A120077(n): n in [2..30]]; // G. C. Greubel, Apr 25 2023

Table[Denominator[HarmonicNumber[n,2] -1/n], {n,2,40}] (* G. C. Greubel, Apr 25 2023 *)

a(n) = denominator(sum(j=1,n-1,1/j^2-1/n^2)) \\ Jeppe Stig Nielsen, Aug 18 2019

a(n) = denominator(sum(j=1,n,1/j^2) - 1/n) \\ Jeppe Stig Nielsen, Aug 18 2019

def A120077(n): return denominator(harmonic_number(n,2) - 1/n)
[A120077(n) for n in range(2,31)] # G. C. Greubel, Apr 25 2023

a(n) = denominator(r(m)), with the rationals r(m) = Sum_{n=1..m-1} A120072(m,n)/A120073(m,n), m >= 2.
The rationals are r(m) = Zeta(2; m-1) - (m-1)/m^2, m>=2, with the partial sums Zeta(2; n) = Sum_{k=1..n} 1/k^2. See the W. Lang link under A103345.
O.g.f. for the rationals r(m), m>=2: log(1-x) + polylog(2,x)/(1-x).

a(21)-a(23) from Jeppe Stig Nielsen, Aug 18 2019

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).

A120076 Numerators of row sums of rational triangle A120072/A120073.

Original entry on oeis.org

3, 37, 169, 4549, 4769, 241481, 989549, 9072541, 1841321, 225467009, 227698469, 38801207261, 39076419341, 196577627041, 790503882349, 229526961468061, 230480866420061, 83512167402400421, 3351610394325821

Offset: 2

Wolfdieter Lang, Jul 20 2006

The corresponding denominators are given by A120077.

See the W. Lang link under A120072 for more details.

			The rationals a(m)/A120077(m), m>=2, begin with (3/4, 37/36, 169/144, 4549/3600, 4769/3600, ...).

G. C. Greubel, Table of n, a(n) for n = 2..1000

Cf. A120070, A120072, A120073, A120074, A120075, A120077.

A120076:= func< n | Numerator( (&+[1/k^2: k in [1..n]]) -1/n) >;
[A120076(n): n in [2..30]]; // G. C. Greubel, Apr 24 2023

Table[Numerator[HarmonicNumber[n,2] -1/n], {n,2,40}] (* G. C. Greubel, Apr 24 2023 *)

def A120076(n): return numerator(harmonic_number(n,2) - 1/n)
[A120076(n) for n in range(2,31)] # G. C. Greubel, Apr 24 2023

a(n) = numerator(r(m)), with the rationals r(m) = Sum_{n=1..m-1} A120072(m,n)/A120073(m,n), m >= 2.
The rationals are r(m) = Zeta(2; m-1) - (m-1)/m^2, m >= 2, with the partial sums Zeta(2; n) = Sum_{k=1..n} 1/k^2. See the W. Lang link in A103345.
O.g.f. for the rationals r(m), m>=2: log(1-x) + polylog(2,x)/(1-x).

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).

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A136682 Numbers k such that A119682(k) is prime.

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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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A291456 a(n) = (n!)^6 * Sum_{i=1..n} 1/i^6.

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A120077 Denominators of row sums of rational triangle A120072/A120073.

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

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