A136677 -id:A136677 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

1, 7, 197, 1549, 195353, 194353, 66879079, 533875007, 14436577189, 14420574181, 19209787242911, 19197460851911, 42198121495296467, 6025866788581781, 6027847576222613, 48209723660000029, 236907853607882606477

Offset: 1

Alexander Adamchuk, Jan 16 2008

a(n) is prime for n in A136683.

Lim_{n -> infinity} a(n)/A334582(n) = A197070. - Petros Hadjicostas, May 07 2020

			The first few fractions are 1, 7/8, 197/216, 1549/1728, 195353/216000, 194353/216000, 66879079/74088000, 533875007/592704000, ... = a(n)/A334582(n). - _Petros Hadjicostas_, May 06 2020

Robert Israel, Table of n, a(n) for n = 1..768
Eric Weisstein's World of Mathematics, Harmonic Number.
Wikipedia, Dirichlet eta function.

Cf. A001008, A007406, A007408, A007410, A058313, A099828, A103345, A119682, A120296, A136676, A136677, A136681, A136682, A136683, A136684, A136685, A136686, A197070, A334582 (denominators).

map(numer,ListTools:-PartialSums([seq((-1)^(k+1)/k^3, k=1..100)])); # Robert Israel, Nov 09 2023

(* Program #1 *) Table[Numerator[Sum[(-1)^(k+1)/k^3, {k,1,n}]], {n,1,50}]
(* Program #2 *) Numerator[Accumulate[Table[(-1)^(k+1) 1/k^3, {k,50}]]] (* Harvey P. Dale, Feb 12 2013 *)

a(n) = numerator(sum(k=1, n, (-1)^(k+1)/k^3)); \\ Michel Marcus, May 07 2020

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

Original entry on oeis.org

1, 31, 7565, 241837, 755989457, 755889457, 12705011703799, 406547611705943, 98792790681344149, 98791774426324117, 15910615688635928566967, 15910549913780913466967, 5907492176026179821253778331

Offset: 1

Alexander Adamchuk, Jan 16 2008

a(n) is prime for n in A136685.

Lim_{n -> infinity} a(n)/A334604(n) = A267316 = (15/16)*A013663. - Petros Hadjicostas, May 07 2020

			The first few fractions are 1, 31/32, 7565/7776, 241837/248832, 755989457/777600000, 755889457/777600000, ... = a(n)/A334604(n). - _Petros Hadjicostas_, May 07 2020

Eric Weisstein's World of Mathematics, Harmonic Number.

Cf. A001008, A007406, A007408, A007410, A013663, A058313, A099828, A103345, A119682, A120296, A136675, A136677, A136681, A136682, A136683, A136684, A136685, A136686, A267316, A334604 (denominators).

Table[ Numerator[ Sum[ (-1)^(k+1)/k^5, {k,1,n} ] ], {n,1,30} ]

a(n) = numerator(sum(k=1, n, (-1)^(k+1)/k^5)); \\ Michel Marcus, May 07 2020

A136681 Numbers k such that A058313(k) is prime.

Original entry on oeis.org

3, 4, 5, 6, 9, 10, 13, 16, 17, 18, 37, 43, 58, 121, 124, 126, 137, 203, 247, 283, 285, 286, 289, 317, 424, 508, 751, 790, 937, 958, 1066, 1097, 1151, 1166, 1194, 1199, 1235, 1414, 1418, 1460, 1498, 1573, 2090, 2122, 2691, 2718, 3030, 3426, 3600, 3653, 3737

Offset: 1

Alexander Adamchuk, Jan 16 2008

nonn

A058313(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j.

Eric Weisstein's World of Mathematics, Harmonic Number

Cf. A058313, A119682, A120296, A136675, A136676, A136677.
Cf. A001008, A007406, A007408, A007410, A099828, A103345.
Cf. A136682, A136683, A136684, A136685, A136686.

Do[ f=Numerator[ Sum[ (-1)^(k+1)*1/k, {k,1,n} ] ]; If[ PrimeQ[f], Print[ {n,f} ] ], {n,1,317} ]

isok(n) = isprime(numerator(sum(k=1, n, (-1)^(k+1)/k))); \\ Michel Marcus, Mar 14 2019

a(25)-a(30) from James R. Buddenhagen, Sep 22 2015

a(31)-a(51) from Amiram Eldar, Mar 14 2019

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

Original entry on oeis.org

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

A275703 Decimal expansion of the Dirichlet eta function at 6.

Original entry on oeis.org

9, 8, 5, 5, 5, 1, 0, 9, 1, 2, 9, 7, 4, 3, 5, 1, 0, 4, 0, 9, 8, 4, 3, 9, 2, 4, 4, 4, 8, 4, 9, 5, 4, 2, 6, 1, 4, 0, 4, 8, 8, 5, 6, 9, 3, 4, 6, 9, 3, 2, 6, 8, 8, 8, 0, 3, 4, 8, 3, 3, 3, 9, 3, 2, 5, 4, 1, 9, 6, 8, 0, 2, 1, 8, 6, 2, 7, 1, 7, 1, 3, 5, 7, 3, 9, 3, 7, 2, 9, 1, 1, 2, 7, 9, 5, 5, 9, 4, 6, 4

Offset: 0

Terry D. Grant, Aug 05 2016

It appears that each sum of a Dirichlet eta function is 1/2^(x-1) less than the zeta(x), where x is a positive integer > 1. In this case, eta(x) = eta(6) = (31/32)*zeta(6) = 31*(Pi^6)/30240. Therefore eta(6) = 1/2^(6-1) or 1/32nd less than zeta(6) (see A013664). [Edited by Petros Hadjicostas, May 07 2020]

			31*(Pi^6)/30240 = 0.9855510912974...

Cf. A002162 (decimal expansion of value at 1), A072691 (value at 2), A197070 (value at 3), A267315 (value at 4), A267316 (value at 5), A275710 (value at 7).

Cf. A013664, A136677, A334605.

Mathematica
```
RealDigits[31*(Pi^6)/30240,10,100]
```

s = RLF(0); s
RealField(110)(s)
for i in range(1, 10000): s -= (-1)^i / i^6
print(s) # Terry D. Grant, Aug 05 2016

eta(6) = 31*(Pi^6)/30240 = 31*A092732/30240 = Sum_{n>=1} (-1)^(n+1)/n^6.

eta(6) = lim_{n -> infinity} A136677(n)/A334605(n). - Petros Hadjicostas, May 07 2020

A334605 Denominator of Sum_{k=1..n} (-1)^(k+1)/k^6.

Original entry on oeis.org

1, 64, 46656, 2985984, 46656000000, 5184000000, 609892416000000, 39033114624000000, 256096265048064000000, 256096265048064000000, 453690155404813307904000000, 453690155404813307904000000, 2189875725319351517910798336000000

Offset: 1

Petros Hadjicostas, May 07 2020

Lim_{n -> infinity} A136677(n)/a(n) = A275703 = (31/32)*A013664.

			The first few fractions are: 1, 63/64, 45991/46656, 2942695/2985984, 45982595359/46656000000, 5109066151/5184000000, ... = A136677/A334605.

Cf. A013664, A136677 (numerators), A275703.

Denominator @ Accumulate[Table[(-1)^(k + 1)/k^6, {k, 1, 13}]] (* Amiram Eldar, May 07 2020 *)

a(n) = denominator(sum(k=1, n, (-1)^(k+1)/k^6)); \\ Michel Marcus, May 07 2020

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

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

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

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A136681 Numbers k such that A058313(k) is prime.

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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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