A035105 -id:A035105 - cp's OEIS frontend

A059248 Numerator of 1/F(1) + 1/F(2) + 1/F(3) + ... + 1/F(n), where F(n) is the n-th Fibonacci number (A000045).

1, 2, 5, 17, 91, 379, 5047, 35849, 614893, 6800951, 607326679, 3651532639, 851897554247, 24724573280923, 301787157353771, 14188276949397301, 22662903194758542865, 430644772287132696121, 1800653989272587268758525

Offset: 1

Avi Peretz (njk(AT)netvision.net.il), Jan 22 2001

			a(4) = 17 because 1/F1 + 1/F2 + 1/F3 + 1/F4 = 1 + 1 + 1/2 + 1/3 = 17/6 and the numerator is 17.
1, 2, 5/2, 17/6, 91/30, 379/120, 5047/1560, 35849/10920, 614893/185640, 6800951/2042040, 607326679/181741560, ... = A059248/A035105.

G. C. Greubel, Table of n, a(n) for n = 1..100
Naim Tuglu, Can Kizilates, Seyhun Kesim, On the Harmonic and Hyperharmonic Fibonacci Numbers, arXiv:1505.04284 [math.NT], 2015 (see Table 1).

Cf. A000045, A035105, Cf. A059246/A059247.

BB:=n->sum(1/fibonacci(i), i=1..n): a:=n->floor(numer(BB(n))): seq(a(n), n=1..19); # Zerinvary Lajos, Mar 28 2007

Table[ 1 / Fibonacci[n], {n, 1, 19}] // Accumulate // Numerator (* Jean-François Alcover, Mar 07 2013 *)

lista(nn) = s = 0; for (n=1, nn, s += 1/fibonacci(n); print1(numerator(s), ", ");); \\ Michel Marcus, Nov 28 2014

More terms from Naohiro Nomoto, Jun 21 2001

Offset changed to 1 by Michel Marcus, Nov 28 2014

A129655 Numbers that set a new record for number of Fibonacci divisors.

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 5040, 55440, 720720, 12252240, 232792560, 6750984240, 276790353840, 12732356276640, 523410559111440, 24076885719126240, 1131613628798933280, 100713612963105061920, 20042008979657907322080

Offset: 1

Jason Earls, May 19 2007

From Donovan Johnson, Jul 07 2009: (Start)
a(15) <= 598420745002080,
a(16) <= 36503665445126880,
a(17) <= 1131613628798933280,
a(18) <= 100713612963105061920. (End)
From Robert Israel, Sep 26 2019: (Start)
a(15) <= 523410559111440,
a(16) <= 24076885719126240. (End)
From David A. Corneth, Sep 27 2019: (Start)
a(19) <= 20042008979657907322080,
a(20) <= 4669788092260292406044640,
a(21) <= 1312210453925142166098543840,
a(22) <= 414821946023574034721351415840,
a(23) <= 116564966832624303756699747851040,
a(24) <= 37417354353272401505900619060183840,
a(25) <= 19494441618054921184574222530355780640,
a(26) <= 31132623264033709131765033380978181682080,
a(27) <= 67277598873576845433744237136293850614974880. (End)
From a(1) up to a(14), last known term, this sequence is equivalent to: a(n) is the smallest number that has exactly n Fibonacci divisors (A000045). The products of the new Fibonacci divisors that appear successively are in A349100. - Bernard Schott, Jul 15 2022

			5040 has 60 divisors with 7 of them being Fibonacci numbers, namely 1, 2, 3, 5, 8, 21 and 144.

J. Earls, Red Zen, Lulu Press, NY, 2006, p. 105.

David A. Corneth, Some actual values and some upper bounds on a(n), for n=1..134
David A. Corneth, PARI program

Cf. A000045, A005086, A035105, A349100.

a(n) <= A035105(n+1). - Daniel Suteu, Sep 27 2019

More terms from Donovan Johnson, Feb 26 2008
a(14) from Donovan Johnson, Jul 07 2009
a(15)-a(19) confirmed by David A. Corneth, Sep 06 2024

A233287 a(n) = lcm_{i=1..n} A001177(i); partial LCM-products of Fibonacci entry points.

Original entry on oeis.org

1, 3, 12, 12, 60, 60, 120, 120, 120, 120, 120, 120, 840, 840, 840, 840, 2520, 2520, 2520, 2520, 2520, 2520, 2520, 2520, 12600, 12600, 12600, 12600, 12600, 12600, 12600, 12600, 12600, 12600, 12600, 12600, 239400, 239400, 239400, 239400, 239400, 239400, 2633400

Offset: 1

Antti Karttunen, Dec 13 2013

nonn

From n=3 onward it seems that lcm_{i=1..n} A001175(i) = 2*a(n).

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences related to lcm's

Records occur at A233282. Cf. also A233283, A233284, A233285, A001175..A001177, A035105.

a(1)=1, and for n > 1, a(n) = lcm(A001177(n), a(n-1)).

a(n) = lcm_{i=1..n} A001177(i). [the least common multiple of all terms from A001177(1) to A001177(n)]

A059247 Denominator of Sum_{j=1..n} d(j)/n, where d = number of divisors function (A000005).

Original entry on oeis.org

1, 2, 3, 1, 1, 3, 7, 2, 9, 10, 11, 12, 13, 14, 1, 8, 17, 9, 19, 10, 3, 11, 23, 2, 25, 2, 27, 28, 29, 10, 31, 32, 11, 34, 35, 9, 37, 19, 13, 20, 41, 1, 43, 1, 45, 23, 1, 8, 49, 50, 51, 52, 53, 54, 5, 56, 19, 58, 59, 20, 61, 62, 3, 8, 65, 33, 67, 17, 69, 35, 71

Offset: 1

N. J. A. Sloane, Jan 21 2001

			1, 3/2, 5/3, 2, 2, 7/3, 16/7, 5/2, 23/9, 27/10, ...

M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 1999; see p. 135.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A006218, A059246 and also A059248/A035105.

Denominator[Table[Sum[DivisorSigma[0, j]/n, {j,1,n}], {n,1,100}]] (* G. C. Greubel, Jan 02 2016 *)

a(n) = denominator(sum(j=1, n, numdiv(j))/n); \\ Michel Marcus, Jan 03 2017

from math import isqrt, gcd
def A059247(n): return n//gcd(n,(lambda m: 2*sum(n//k for k in range(1, m+1))-m*m)(isqrt(n))) # Chai Wah Wu, Oct 08 2021

a(n) = denominator(A006218(n)/n). - Michel Marcus, Jan 03 2017

A062954 Least common multiple of Lucas numbers L(0), L(1), ..., L(n).

Original entry on oeis.org

2, 2, 6, 12, 84, 924, 2772, 80388, 3778236, 71786484, 2943245844, 585705922956, 13471236227988, 7018514074781748, 1972202455013671188, 61138276105423806828, 134932175364670341669396, 481842798227237790101413116, 154671538230943330622553610236

Offset: 0

Reiner Martin, Jul 21 2001

Cf. A000032, A035105.

Module[{nn=20,luc},luc=LucasL[Range[0,nn]];Rest[Table[LCM@@Take[luc,n],{n,nn}]]] (* Harvey P. Dale, Jun 29 2015 *)

A000032(n) = fibonacci(n+1)+fibonacci(n-1)
a(n) = {v = 1; for (i=0, n, v = lcm(v, A000032(i));); return (v);} \\ Michel Marcus, Jul 22 2013

a(0) and more terms from Sean A. Irvine, Apr 15 2023

A292794 Numbers not congruent to A000045(k) mod A000045(k+1) for all k > 1.

Original entry on oeis.org

0, 4, 6, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 64, 66, 70, 72, 82, 84, 90, 94, 96, 100, 102, 106, 114, 120, 124, 126, 130, 132, 136, 142, 150, 154, 156, 162, 166, 172, 174, 180, 184, 186, 192, 196, 204, 210, 214, 220, 222, 226, 232, 234, 240, 246, 250, 252, 256

Offset: 0

Ely Golden, Sep 23 2017

For n > 0, also numbers n such that A292032(n) = 1.
It is conjectured that A035105(n) is always a member of this sequence for n >= 4 but this remains unproved.
This is the complement of (1 + 2Z) U (2 + 3Z) U (3 + 5Z) U (5 + 8Z) U ..., see also the Example section. - M. F. Hasler, Feb 25 2018

			a(2) = 6 since 6 mod 2 = 0, 6 mod 3 = 0, 6 mod 5 = 1, and 6 mod 8 = 6. (No other terms of A000045 need to be checked since the "illegal congruences" are all greater than 6, yet 6 is always congruent to 6 for those terms.)
From _M. F. Hasler_, Feb 26 2018: (Start)
This set can be constructed using a sieve which removes:
- first all numbers == 1 (mod 2), there remain the even numbers 0, 2, 4...;
- then all numbers == 2 (mod 3), i.e., == 2 (mod 6), there remain the numbers == 0 or 4 (mod 6): 0, 4, 6, 10, 12, 16, 18, 22, 24, 28, ...;
- then all numbers == 3 (mod 5), i.e., == 8 (mod 10), these are the numbers == 18 or 28 (mod 30), there remain numbers == 0, 4, 6, 10, 12, 16, 22 or 24 (mod 30);
- then all those == 5 (mod 8), but all these are odd;
- then all those == 8 (mod 13), i.e., == 8 (mod 26): there are 8 of these in [1..30*13], and there remain 8*(13-1) residue classes mod 30*13.
- then all those == 13 (mod 21): there are 48 of these left in [1..30*13*7], and there remain 8*12*7-48 = 48*(14-1) residue classes mod 30*13*7.
- then again there are none to remove == 21 (mod 34);
- then those == 34 (mod 55): these are 12*13 of the remaining 48*13*11 residue classes mod 30*13*7*11, so there remain 12*13*(4*11-1) of these; and so on.
This yields as upper bound of the asymptotic density: 1/2 * 2/3 * 4/5 * 12/13 * 13*14 * 43/44 ~ 0.223, the actual value is 0.2187...
(End)

Ely Golden, Table of n, a(n) for n = 0..10000
Ely Golden, Python program for generating terms of this sequence

Cf. A292030, A292031, A292032.

Cf. A300004 for the sequence of first differences.

{0}~Join~Select[Range[3, 250], Function[n, NoneTrue[Block[{k = {1, 1}}, While[Last@ k <= n, AppendTo[k, Total@ Take[k, -2]]]; Partition[Most@ k, 2, 1]], Mod[n, #2] == #1 & @@ # &]]] (* Michael De Vlieger, Mar 19 2018 *)

is_A292794(n,F=1)=!for(k=3,oo,F==n%(F=fibonacci(k))&&return;F>n&&break) \\ M. F. Hasler, Feb 25 2018

a(10^7) = 45721410, a(10^8) = 457214230, a(10^9) = 4572142416. - Jacques Tramu, Feb 26 2018

A180402 a(n) = lcm(1,...,Fibonacci(n)).

Original entry on oeis.org

1, 1, 2, 6, 60, 840, 360360, 232792560, 144403552893600, 164249358725037825439200, 718766754945489455304472257065075294400, 33312720618553145840562713089120360606823375590405920630576000

Offset: 1

Vaclav Kotesovec, Sep 02 2010

nonn

Also least period for number of ways of placing k non-attacking queens on an n X n chessboard. [conjectured by Kotesovec; proved for n <= 5. - Thomas Zaslavsky, Jun 24 2018]

Alois P. Heinz, Table of n, a(n) for n = 1..17
Christopher R. H. Hanusa, T. Zaslavsky, S. Chaiken, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv preprint arXiv:1609.00853 [math.CO], 2016. See Table 8.1.
V. Kotesovec, Non-attacking chess pieces, 6ed, p.31, 2013

Cf. A178717, A178721, A078491, A218492, A035105, A062954.

a:= n-> ilcm($1..(<<0|1>, <1|1>>^n)[1,2]):
seq(a(n), n=1..14);  # Alois P. Heinz, Aug 12 2017

Table[Apply[LCM, Range[Fibonacci[k]]], {k, 1, 10}]
Array[LCM @@ Range@Fibonacci@# &, 12] (* Robert G. Wilson v, Sep 05 2010 *)

a(n) = lcm([1..fibonacci(n)]); \\ Michel Marcus, Jun 24 2018

a(11) onwards from Robert G. Wilson v, Sep 05 2010

A218492 a(n) = lcm(1,...,L(n)), where L(n) = n-th Lucas number.

Original entry on oeis.org

2, 1, 6, 12, 420, 27720, 12252240, 2329089562800, 442720643463713815200, 410555180440430163438262940577600, 10514768575588513054648621420819083891762891880353600, 337293588832926264639465766794841407432394382785157234228847021917234018060677390066992000

Offset: 0

Vaclav Kotesovec, Oct 30 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..16

Cf. A000032, A180402, A062954, A035105.

Table[Apply[LCM, Range[LucasL[k]]], {k, 0, 10}]

A225904 Numerator of Sum_{k=1..n} 1/L(k) where L(n) is the n-th Lucas number (A000204).

Original entry on oeis.org

1, 4, 19, 145, 1679, 5191, 153311, 7286005, 69689327, 2869226821, 572447760301, 6593608277800, 3438637721790797, 966842075996112436, 119933240206586434591, 264753799412041684949165, 945570749875765527295137611, 303554979754466691916744807193

Offset: 1

Michel Lagneau, May 20 2013

			1, 4/3, 19/12, 145/84, 1679/924, 5191/2772, ... = A225904/A225905.

Cf. A000204, A059248, A035105, A225905.

Numerator[Table[Sum[1/LucasL[k], {k, 1, n}], {n, 1, 30}]]
Numerator[Accumulate[1/LucasL[Range[20]]]] (* Harvey P. Dale, Sep 16 2016 *)

A225905 Denominator of Sum_{k=1..n} 1/L(k) where L(n) is the n-th Lucas number (A000204).

Original entry on oeis.org

1, 3, 12, 84, 924, 2772, 80388, 3778236, 35893242, 1471622922, 292852961478, 3367809056997, 1754628518695437, 493050613753417797, 61138276105423806828, 134932175364670341669396, 481842798227237790101413116, 154671538230943330622553610236

Offset: 1

Michel Lagneau, May 20 2013

			1, 4/3, 19/12, 145/84, 1679/924, 5191/2772, ... = A225904/A225905.

Cf. A000204, A059248, A035105, A225904.

Denominator[Table[Sum[1/LucasL[k], {k, 1, n}], {n, 1, 30}]]
Denominator[Accumulate[1/LucasL[Range[20]]]] (* Harvey P. Dale, Mar 11 2015 *)

cp's OEIS Frontend

A059248 Numerator of 1/F(1) + 1/F(2) + 1/F(3) + ... + 1/F(n), where F(n) is the n-th Fibonacci number (A000045).

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A129655 Numbers that set a new record for number of Fibonacci divisors.

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A233287 a(n) = lcm_{i=1..n} A001177(i); partial LCM-products of Fibonacci entry points.

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A059247 Denominator of Sum_{j=1..n} d(j)/n, where d = number of divisors function (A000005).

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A062954 Least common multiple of Lucas numbers L(0), L(1), ..., L(n).

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A292794 Numbers not congruent to A000045(k) mod A000045(k+1) for all k > 1.

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Mathematica

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A180402 a(n) = lcm(1,...,Fibonacci(n)).

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Mathematica

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A218492 a(n) = lcm(1,...,L(n)), where L(n) = n-th Lucas number.

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