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A111035
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Numbers n that divide the sum of the first n nonzero Fibonacci numbers.
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11
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1, 2, 24, 48, 72, 77, 96, 120, 144, 192, 216, 240, 288, 319, 323, 336, 360, 384, 432, 480, 576, 600, 648, 672, 720, 768, 864, 960, 1008, 1080, 1104, 1152, 1200, 1224, 1296, 1320, 1344, 1368, 1440, 1517, 1536, 1680, 1728, 1800, 1920, 1944, 2016, 2064, 2160
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OFFSET
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1,2
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COMMENTS
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The sum of the first n nonzero Fibonacci numbers is F(n+2)-1, sequence A000071. Knott discusses the factorization of these numbers. Most of the terms are divisible by 24. - T. D. Noe, Oct 10 2005, edited by M. F. Hasler, Mar 01 2020
All terms are either multiples of 24 (cf. A124455) or odd (cf. A331976) or congruent to 2 (mod 12), cf. A331870 where this statement is proved. - M. F. Hasler, Mar 01 2020
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LINKS
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Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Ron Knott, The Mathematical Magic of the Fibonacci Numbers
Daniel Yaqubi and Amirali Fatehizadeh, Some results on average of Fibonacci and Lucas sequences, arXiv:2001.11839 [math.CO], 2020.
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FORMULA
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{n: n| A000071(n+2)}. - R. J. Mathar, Feb 05 2020
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EXAMPLE
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2 | 4, 24 | 121392, 48 | 12586269024, ... [Corrected by M. F. Hasler, Feb 06 2020]
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MAPLE
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select(n-> irem(combinat[fibonacci](n+2)-1, n)=0, [$1..3000])[]; # G. C. Greubel, Feb 03 2020
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MATHEMATICA
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Select[Range[3000], Mod[Fibonacci[ #+2]-1, # ]==0&] (* T. D. Noe, Oct 06 2005 *)
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PROG
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(PARI) is(n)=((Mod([1, 1; 1, 0], n))^(n+2))[1, 2]==1 \\ Charles R Greathouse IV, Feb 04 2013
(Magma) [1] cat [n: n in [1..3000] | Fibonacci(n+2) mod n eq 1 ]; // G. C. Greubel, Feb 03 2020
(Sage) [n for n in (1..3000) if mod(fibonacci(n+2), n)==1 ] # G. C. Greubel, Feb 03 2020
(GAP) Filtered([1..3000], n-> ((Fibonacci(n+2)-1) mod n)=0 ); # G. C. Greubel, Feb 03 2020
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CROSSREFS
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See A101907 for another version.
Cf. A111058 (the analog for Lucas numbers).
Cf. A124455 (k for a(n) = 24k), A124456 (other a(n)), A331976 (odd a(n)), A331870 (even a(n) != 24k).
Sequence in context: A119070 A181283 A073215 * A349188 A249277 A002552
Adjacent sequences: A111032 A111033 A111034 * A111036 A111037 A111038
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KEYWORD
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nonn
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AUTHOR
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Joseph L. Pe, Oct 05 2005
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EXTENSIONS
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More terms from Rick L. Shepherd and T. D. Noe, Oct 06 2005
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STATUS
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approved
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