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Search: a136677 -id:a136677
Displaying 1-10 of 10 results found. page 1
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A136686 Numbers k such that A136677(k) is prime. +20
8
19, 47, 164, 235, 504, 1109, 1112, 5134, 9222 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A136677(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j^6.
LINKS
Table of n, a(n) for n=1..9.
Eric Weisstein's World of Mathematics, Harmonic Number.
MATHEMATICA
Do[ f=Numerator[ Sum[ (-1)^(k+1)*1/k^6, {k, 1, n} ] ]; If[ PrimeQ[f], Print[ {n, f} ] ], {n, 1, 130} ]
CROSSREFS
Cf. A058313, A119682, A136675, A120296, A136676, A136677. Cf. A001008, A007406, A007408, A007410, A099828, A103345. Cf. A136681, A136682, A136683, A136684, A136685.
KEYWORD
nonn,hard,more
AUTHOR
Alexander Adamchuk, Jan 16 2008
EXTENSIONS
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
STATUS
approved
A136675 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^3. +10
11
1, 7, 197, 1549, 195353, 194353, 66879079, 533875007, 14436577189, 14420574181, 19209787242911, 19197460851911, 42198121495296467, 6025866788581781, 6027847576222613, 48209723660000029, 236907853607882606477 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
a(n) is prime for n in A136683.
Lim_{n -> infinity} a(n)/A334582(n) = A197070. - Petros Hadjicostas, May 07 2020
LINKS
Robert Israel, Table of n, a(n) for n = 1..768
Eric Weisstein's World of Mathematics, Harmonic Number.
Wikipedia, Dirichlet eta function.
EXAMPLE
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
MAPLE
map(numer, ListTools:-PartialSums([seq((-1)^(k+1)/k^3, k=1..100)])); # Robert Israel, Nov 09 2023
MATHEMATICA
(* 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 *)
PROG
(PARI) a(n) = numerator(sum(k=1, n, (-1)^(k+1)/k^3)); \\ Michel Marcus, May 07 2020
CROSSREFS
Cf. A001008, A007406, A007408, A007410, A058313, A099828, A103345, A119682, A120296, A136676, A136677, A136681, A136682, A136683, A136684, A136685, A136686, A197070, A334582 (denominators).
KEYWORD
frac,nonn
AUTHOR
Alexander Adamchuk, Jan 16 2008
STATUS
approved
A136676 Numerator of Sum_{k=1..n} (-1)^(k+1)/k^5. +10
10
1, 31, 7565, 241837, 755989457, 755889457, 12705011703799, 406547611705943, 98792790681344149, 98791774426324117, 15910615688635928566967, 15910549913780913466967, 5907492176026179821253778331 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
a(n) is prime for n in A136685.
Lim_{n -> infinity} a(n)/A334604(n) = A267316 = (15/16)*A013663. - Petros Hadjicostas, May 07 2020
LINKS
Table of n, a(n) for n=1..13.
Eric Weisstein's World of Mathematics, Harmonic Number.
EXAMPLE
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
MATHEMATICA
Table[ Numerator[ Sum[ (-1)^(k+1)/k^5, {k, 1, n} ] ], {n, 1, 30} ]
PROG
(PARI) a(n) = numerator(sum(k=1, n, (-1)^(k+1)/k^5)); \\ Michel Marcus, May 07 2020
CROSSREFS
Cf. A001008, A007406, A007408, A007410, A013663, A058313, A099828, A103345, A119682, A120296, A136675, A136677, A136681, A136682, A136683, A136684, A136685, A136686, A267316, A334604 (denominators).
KEYWORD
frac,nonn
AUTHOR
Alexander Adamchuk, Jan 16 2008
STATUS
approved
A136681 Numbers k such that A058313(k) is prime. +10
8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A058313(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j.
LINKS
Table of n, a(n) for n=1..51.
Eric Weisstein's World of Mathematics, Harmonic Number
MATHEMATICA
Do[ f=Numerator[ Sum[ (-1)^(k+1)*1/k, {k, 1, n} ] ]; If[ PrimeQ[f], Print[ {n, f} ] ], {n, 1, 317} ]
PROG
(PARI) isok(n) = isprime(numerator(sum(k=1, n, (-1)^(k+1)/k))); \\ Michel Marcus, Mar 14 2019
CROSSREFS
Cf. A058313, A119682, A120296, A136675, A136676, A136677.
Cf. A001008, A007406, A007408, A007410, A099828, A103345.
Cf. A136682, A136683, A136684, A136685, A136686.
KEYWORD
nonn
AUTHOR
Alexander Adamchuk, Jan 16 2008
EXTENSIONS
a(25)-a(30) from James R. Buddenhagen, Sep 22 2015
a(31)-a(51) from Amiram Eldar, Mar 14 2019
STATUS
approved
A136682 Numbers k such that A119682(k) is prime. +10
8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A119682(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j^2.
LINKS
Table of n, a(n) for n=1..45.
Eric Weisstein's World of Mathematics, Harmonic Number
MATHEMATICA
Do[ f=Numerator[ Sum[ (-1)^(k+1)*1/k^2, {k, 1, n} ] ]; If[ PrimeQ[f], Print[ {n, f} ] ], {n, 1, 125} ]
CROSSREFS
Cf. A058313, A119682, A120296, A136675, A136676, A136677.
Cf. A001008, A007406, A007408, A007410, A099828, A103345.
Cf. A136681, A136683, A136684, A136685, A136686.
KEYWORD
nonn
AUTHOR
Alexander Adamchuk, Jan 16 2008
EXTENSIONS
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
STATUS
approved
A136683 Numbers k such that A136675(k) is prime. +10
8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A136675(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j^3.
LINKS
Table of n, a(n) for n=1..32.
Eric Weisstein's World of Mathematics, Harmonic Number
MATHEMATICA
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 *)
PROG
(PARI) isok(n) = ispseudoprime(numerator(sum(k=1, n, (-1)^(k+1) / k^3))); \\ Daniel Suteu, Mar 15 2019
CROSSREFS
Cf. A058313, A119682, A120296, A136675, A136676, A136677.
Cf. A001008, A007406, A007408, A007410, A099828, A103345.
Cf. A136681, A136682, A136684, A136685, A136686.
KEYWORD
nonn,more
AUTHOR
Alexander Adamchuk, Jan 16 2008
EXTENSIONS
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
STATUS
approved
A136684 Numbers k such that A120296(k) is prime. +10
8
3, 5, 8, 11, 20, 38, 61, 65, 71, 80, 83, 93, 233, 704, 1649, 2909, 3417, 3634, 9371 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A120296(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j^4.
LINKS
Table of n, a(n) for n=1..19.
Eric Weisstein's World of Mathematics, Harmonic Number
MATHEMATICA
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 *)
CROSSREFS
Cf. A058313, A119682, A136675, A120296, A136676, A136677.
Cf. A001008, A007406, A007408, A007410, A099828, A103345.
Cf. A136681, A136682, A136683, A136685, A136686.
KEYWORD
nonn,more
AUTHOR
Alexander Adamchuk, Jan 16 2008
EXTENSIONS
More terms from Harvey P. Dale, Aug 28 2012
a(15)-a(19) from Robert Price, Apr 23 2019
STATUS
approved
A136685 Numbers k such that A136676(k) is prime. +10
8
2, 19, 51, 78, 84, 130, 294, 910, 2223, 2642, 3261, 4312, 4973, 7846, 9439 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A136676(k) = Numerator of Sum_{j=1..k} (-1)^(j+1)/j^5.
LINKS
Table of n, a(n) for n=1..15.
Eric Weisstein's World of Mathematics, Harmonic Number
MATHEMATICA
Do[ f=Numerator[ Sum[ (-1)^(k+1)*1/k^5, {k, 1, n} ] ]; If[ PrimeQ[f], Print[ {n, f} ] ], {n, 1, 130} ]
CROSSREFS
Cf. A058313, A119682, A136675, A120296, A136676, A136677.
Cf. A001008, A007406, A007408, A007410, A099828, A103345.
Cf. A136681, A136682, A136683, A136684, A136686.
KEYWORD
nonn,more
AUTHOR
Alexander Adamchuk, Jan 16 2008
EXTENSIONS
a(7)-a(8) from Amiram Eldar, Mar 14 2019
a(9)-a(15) from Robert Price, Apr 16 2019
STATUS
approved
A275703 Decimal expansion of the Dirichlet eta function at 6. +10
7
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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]
LINKS
Table of n, a(n) for n=0..99.
FORMULA
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
EXAMPLE
31*(Pi^6)/30240 = 0.9855510912974...
MATHEMATICA
RealDigits[31*(Pi^6)/30240, 10, 100]
PROG
(Sage) 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
CROSSREFS
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.
KEYWORD
nonn,cons
AUTHOR
Terry D. Grant, Aug 05 2016
STATUS
approved
A334605 Denominator of Sum_{k=1..n} (-1)^(k+1)/k^6. +10
3
1, 64, 46656, 2985984, 46656000000, 5184000000, 609892416000000, 39033114624000000, 256096265048064000000, 256096265048064000000, 453690155404813307904000000, 453690155404813307904000000, 2189875725319351517910798336000000 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Lim_{n -> infinity} A136677(n)/a(n) = A275703 = (31/32)*A013664.
LINKS
Table of n, a(n) for n=1..13.
EXAMPLE
The first few fractions are: 1, 63/64, 45991/46656, 2942695/2985984, 45982595359/46656000000, 5109066151/5184000000, ... = A136677/A334605.
MATHEMATICA
Denominator @ Accumulate[Table[(-1)^(k + 1)/k^6, {k, 1, 13}]] (* Amiram Eldar, May 07 2020 *)
PROG
(PARI) a(n) = denominator(sum(k=1, n, (-1)^(k+1)/k^6)); \\ Michel Marcus, May 07 2020
CROSSREFS
Cf. A013664, A136677 (numerators), A275703.
KEYWORD
nonn,frac
AUTHOR
Petros Hadjicostas, May 07 2020
STATUS
approved
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Last modified August 29 21:34 EDT 2024. Contains 375518 sequences. (Running on oeis4.)