Search: a136677 -id:a136677
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A334605
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Denominator of Sum_{k=1..n} (-1)^(k+1)/k^6.
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+0
3
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1, 64, 46656, 2985984, 46656000000, 5184000000, 609892416000000, 39033114624000000, 256096265048064000000, 256096265048064000000, 453690155404813307904000000, 453690155404813307904000000, 2189875725319351517910798336000000
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OFFSET
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1,2
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COMMENTS
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LINKS
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EXAMPLE
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The first few fractions are: 1, 63/64, 45991/46656, 2942695/2985984, 45982595359/46656000000, 5109066151/5184000000, ... = A136677/A334605.
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MATHEMATICA
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Denominator @ Accumulate[Table[(-1)^(k + 1)/k^6, {k, 1, 13}]] (* Amiram Eldar, May 07 2020 *)
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PROG
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(PARI) a(n) = denominator(sum(k=1, n, (-1)^(k+1)/k^6)); \\ Michel Marcus, May 07 2020
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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A275703
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Decimal expansion of the Dirichlet eta function at 6.
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+0
7
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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
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OFFSET
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0,1
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COMMENTS
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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]
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LINKS
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FORMULA
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eta(6) = 31*(Pi^6)/30240 = 31*A092732/30240 = Sum_{n>=1} (-1)^(n+1)/n^6.
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EXAMPLE
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31*(Pi^6)/30240 = 0.9855510912974...
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MATHEMATICA
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RealDigits[31*(Pi^6)/30240, 10, 100]
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PROG
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(Sage) s = RLF(0); s
RealField(110)(s)
for i in range(1, 10000): s -= (-1)^i / i^6
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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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
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OFFSET
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1,1
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COMMENTS
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A058313(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j.
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LINKS
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MATHEMATICA
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Do[ f=Numerator[ Sum[ (-1)^(k+1)*1/k, {k, 1, n} ] ]; If[ PrimeQ[f], Print[ {n, f} ] ], {n, 1, 317} ]
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PROG
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(PARI) isok(n) = isprime(numerator(sum(k=1, n, (-1)^(k+1)/k))); \\ Michel Marcus, Mar 14 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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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
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OFFSET
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1,1
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COMMENTS
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A119682(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j^2.
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LINKS
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MATHEMATICA
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Do[ f=Numerator[ Sum[ (-1)^(k+1)*1/k^2, {k, 1, n} ] ]; If[ PrimeQ[f], Print[ {n, f} ] ], {n, 1, 125} ]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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2, 19, 51, 78, 84, 130, 294, 910, 2223, 2642, 3261, 4312, 4973, 7846, 9439
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OFFSET
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1,1
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COMMENTS
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A136676(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j^5.
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LINKS
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MATHEMATICA
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Do[ f=Numerator[ Sum[ (-1)^(k+1)*1/k^5, {k, 1, n} ] ]; If[ PrimeQ[f], Print[ {n, f} ] ], {n, 1, 130} ]
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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3, 5, 8, 11, 20, 38, 61, 65, 71, 80, 83, 93, 233, 704, 1649, 2909, 3417, 3634, 9371
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OFFSET
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1,1
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COMMENTS
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A120296(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j^4.
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LINKS
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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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
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OFFSET
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1,1
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COMMENTS
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A136675(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j^3.
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LINKS
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MATHEMATICA
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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 *)
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PROG
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(PARI) isok(n) = ispseudoprime(numerator(sum(k=1, n, (-1)^(k+1) / k^3))); \\ Daniel Suteu, Mar 15 2019
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A136676
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Numerator of Sum_{k=1..n} (-1)^(k+1)/k^5.
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+0
10
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1, 31, 7565, 241837, 755989457, 755889457, 12705011703799, 406547611705943, 98792790681344149, 98791774426324117, 15910615688635928566967, 15910549913780913466967, 5907492176026179821253778331
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OFFSET
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1,2
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COMMENTS
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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Table[ Numerator[ Sum[ (-1)^(k+1)/k^5, {k, 1, n} ] ], {n, 1, 30} ]
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PROG
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(PARI) a(n) = numerator(sum(k=1, n, (-1)^(k+1)/k^5)); \\ Michel Marcus, May 07 2020
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CROSSREFS
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Cf. A001008, A007406, A007408, A007410, A013663, A058313, A099828, A103345, A119682, A120296, A136675, A136677, A136681, A136682, A136683, A136684, A136685, A136686, A267316, A334604 (denominators).
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KEYWORD
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frac,nonn
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AUTHOR
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STATUS
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approved
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OFFSET
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1,1
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COMMENTS
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A136677(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j^6.
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LINKS
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MATHEMATICA
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Do[ f=Numerator[ Sum[ (-1)^(k+1)*1/k^6, {k, 1, n} ] ]; If[ PrimeQ[f], Print[ {n, f} ] ], {n, 1, 130} ]
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CROSSREFS
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Cf. A058313, A119682, A136675, A120296, A136676, A136677. Cf. A001008, A007406, A007408, A007410, A099828, A103345. Cf. A136681, A136682, A136683, A136684, A136685.
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A136675
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Numerator of Sum_{k=1..n} (-1)^(k+1)/k^3.
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+0
11
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1, 7, 197, 1549, 195353, 194353, 66879079, 533875007, 14436577189, 14420574181, 19209787242911, 19197460851911, 42198121495296467, 6025866788581781, 6027847576222613, 48209723660000029, 236907853607882606477
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OFFSET
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1,2
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COMMENTS
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LINKS
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EXAMPLE
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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
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MAPLE
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map(numer, ListTools:-PartialSums([seq((-1)^(k+1)/k^3, k=1..100)])); # Robert Israel, Nov 09 2023
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MATHEMATICA
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(* 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 *)
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PROG
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(PARI) a(n) = numerator(sum(k=1, n, (-1)^(k+1)/k^3)); \\ Michel Marcus, May 07 2020
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CROSSREFS
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Cf. A001008, A007406, A007408, A007410, A058313, A099828, A103345, A119682, A120296, A136676, A136677, A136681, A136682, A136683, A136684, A136685, A136686, A197070, A334582 (denominators).
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KEYWORD
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frac,nonn
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AUTHOR
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STATUS
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approved
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