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A134162 Let S(k) be the sequence s() defined by s(1) = k; for i>1, s(i) = s(i-1) + gcd(s(i-1), i). Start with the list of natural numbers and remove any k's for which S(k) merges with an S(m) with m < k. This sequence gives conjectural values for the remaining k's. +0
15
1, 2, 4, 8, 16, 20, 44, 92, 110, 136, 152, 170, 172, 188, 200, 212, 236, 242, 256, 272, 316, 332, 368, 440, 488, 500, 590, 616, 620, 632, 650, 676, 704, 710, 742, 788, 824, 848, 892, 946, 952, 968, 1010, 1034, 1036, 1052, 1058, 1088, 1118 (list; graph; listen)
OFFSET

1,2
COMMENT

In other words, these are conjectural initial values k for which the sequences S(k) never merge. The resulting S(k) have been checked to be distinct for 2^60 terms, but it is possible that they merge later on.

CROSSREFS

Cf. A084662, A084663, A106108 and other sequences mentioned in A106108.

Sequence in context: A166156 A089473 A118021 this_sequence A045776 A102252 A001856

Adjacent sequences: A134159 A134160 A134161 this_sequence A134163 A134164 A134165

KEYWORD

nonn

AUTHOR

Eric Rowland (erowland(AT)math.rutgers.edu), Jan 29 2008

A094604 Largest number (up to that point) of consecutive rightmost black cells in the rows of Rule 30 (begun from an initial black cell). a(n)==b(2^n), where b(m) is sequence A094603. +0
7
1, 3, 4, 6, 7, 9, 15, 16, 24, 25, 27, 29, 34, 36, 37, 39, 41, 43, 48, 49, 51, 54, 55, 58, 60, 63, 64, 66, 69, 70, 72, 74, 77, 79, 80, 82, 84, 86, 90, 91, 93, 100 (list; graph; listen)
OFFSET

0,2
COMMENT

The natural number n appears a(n)-a(n-1) times in A094606.

REFERENCES

Wolfram, Stephen. A New Kind of Science. Wolfram Media, 2002.

LINKS

Eric Weisstein's World of Mathematics, Rule 30

CROSSREFS

Cf. A094603, A094605, A094606.

Sequence in context: A137673 A066271 A127594 this_sequence A108654 A131530 A102093

Adjacent sequences: A094601 A094602 A094603 this_sequence A094605 A094606 A094607

KEYWORD

nonn

AUTHOR

Eric S Rowland (erowland(AT)math.rutgers.edu), May 13 2004; revised Aug 10 2005

EXTENSIONS

More terms from Eric S Rowland (erowland(AT)math.rutgers.edu), Jan 21 2006

A094395 Odd composite n such that n divides Fibonacci(n) + 1. +0
6
5777, 10877, 17261, 75077, 80189, 100127, 113573, 120581, 161027, 162133, 163059, 231703, 300847, 430127, 618449, 635627, 667589, 851927, 1033997, 1106327, 1256293, 1388903, 1697183, 1842581, 2263127, 2435423, 2512889, 2662277 (list; graph; listen)
OFFSET

1,1
MATHEMATICA

Select[ Range[3, 300000, 2], !PrimeQ[ # ] && Mod[Fibonacci[ # ] + 1, # ] == 0 &]

CROSSREFS

Sequence in context: A096399 A071132 A094063 this_sequence A094411 A004933 A031664

Adjacent sequences: A094392 A094393 A094394 this_sequence A094396 A094397 A094398

KEYWORD

nonn

AUTHOR

Eric S Rowland (erowland(AT)math.rutgers.edu), May 01 2004

EXTENSIONS

a(6) - a(14) from Robert G. Wilson v (rgwv(AT)rgwv.com), May 1 2004

More terms from Ryan Propper (rpropper(AT)stanford.edu), Aug 03 2005

A094401 Composite n such that n divides both Fibonacci(n-1) and Fibonacci(n) - 1. +0
6
2737, 4181, 6721, 13201, 15251, 34561, 51841, 64079, 64681, 67861, 68251, 90061, 96049, 97921, 118441, 146611, 163081, 179697, 186961, 194833, 197209, 219781, 252601, 254321, 257761, 268801, 272611, 283361, 302101, 303101, 327313, 330929 (list; graph; listen)
OFFSET

1,1
COMMENT

Composite n such that Q^(n-1) = I (mod n), where Q is the Fibonacci matrix {{1,1},{1,0}} and I is the identity matrix. The identity is also true for the primes congruent to 1 or 4 (mod 5), which is sequence A045468. The period of Q^k (mod n) is the same as the period of the Fibonacci numbers F(k) (mod n), A001175. Hence the terms in this sequence are the composite n such that A001175(n) divides n-1. [From T. D. Noe (noe(AT)sspectra.com), Jan 09 2009]

LINKS

T. D. Noe, Table of n, a(n) for n=1..200

Eric Weisstein, MathWorld: Fibonacci Q-Matrix, [From T. D. Noe (noe(AT)sspectra.com), Jan 09 2009]

MATHEMATICA

Select[Range[2, 50000], ! PrimeQ[ # ] && Mod[Fibonacci[ # - 1], # ] == 0 && Mod[Lucas[ # ] - 1, # ] == 0 &]

CROSSREFS

Cf. A005845, A069106, A094394, A094400.

Sequence in context: A145647 A045155 A122473 this_sequence A035774 A107570 A094497

Adjacent sequences: A094398 A094399 A094400 this_sequence A094402 A094403 A094404

KEYWORD

nonn

AUTHOR

Eric S Rowland (erowland(AT)math.rutgers.edu), May 01 2004

EXTENSIONS

More terms from Ryan Propper (rpropper(AT)stanford.edu), Sep 24 2005

A103391 'Even' fractal sequence for the natural numbers: Deleting every even-index term results in the same sequence. +0
6
1, 2, 2, 3, 2, 4, 3, 5, 2, 6, 4, 7, 3, 8, 5, 9, 2, 10, 6, 11, 4, 12, 7, 13, 3, 14, 8, 15, 5, 16, 9, 17, 2, 18, 10, 19, 6, 20, 11, 21, 4, 22, 12, 23, 7, 24, 13, 25, 3, 26, 14, 27, 8, 28, 15, 29, 5, 30, 16, 31, 9, 32, 17, 33, 2, 34, 18, 35, 10, 36, 19, 37, 6, 38, 20, 39, 11, 40, 21, 41, 4, 42 (list; graph; listen)
OFFSET

1,2
COMMENT

A003602 is the 'odd' fractal sequence for the natural numbers.

FORMULA

For n>2, a(n)=A003602(n-2)+1 - Benoit Cloitre (benoit7848c(AT)orange.fr), May 26 2007

CROSSREFS

Cf. A003602.

Sequence in context: A106737 A062821 A162897 this_sequence A061889 A051693 A115980

Adjacent sequences: A103388 A103389 A103390 this_sequence A103392 A103393 A103394

KEYWORD

easy,nonn

AUTHOR

Eric S Rowland (erowland(AT)math.rutgers.edu), Mar 20 2005

A114496 Sum of binomial(n,k)*binomial(2n+k,k) over all k. +0
6
1, 4, 26, 190, 1462, 11584, 93536, 765314, 6323270, 52638760, 440815036, 3709445084, 31340292076, 265683004240, 2258793820988, 19251776923210, 164440378882630, 1407266585304760, 12063701803046300, 103571977632247076 (list; graph; listen)
OFFSET

0,2
COMMENT

Modification of A001850 inspired by the Ap\'ery numbers A005259.

Contribution from Paul Barry (pbarry(AT)wit.ie), Feb 17 2009: (Start)

Central coefficient of (1+4x+5x^2+2x^3)^n. The coefficients are the 4th row of A029635.

The third row of A029635 corresponds to the central Delannoy numbers A001850. (End)

FORMULA

Sum(binomial(n, k)*binomial(2n+k, k), k=0..n)

MATHEMATICA

Sum[Binomial[n, k]*Binomial[2n+k, k], {k, 0, n}]

CROSSREFS

Cf. A114497, A114498.

Cf.: A156886. [From Paul Barry (pbarry(AT)wit.ie), Feb 17 2009]

Sequence in context: A107649 A052763 A084211 this_sequence A127086 A141381 A118971

Adjacent sequences: A114493 A114494 A114495 this_sequence A114497 A114498 A114499

KEYWORD

easy,nonn

AUTHOR

Eric S Rowland (erowland(AT)math.rutgers.edu), Dec 01 2005

A094394 Odd composite n such that n divides Fibonacci(n)-1. +0
5
323, 2737, 4181, 6479, 6721, 7743, 11663, 13201, 15251, 18407, 19043, 23407, 27071, 34561, 34943, 35207, 39203, 44099, 47519, 51841, 51983, 53663, 54839, 64079, 64681, 65471, 67861, 68251, 72831, 78089, 79547, 82983, 86063, 90061, 94667 (list; graph; listen)
OFFSET

0,1
MATHEMATICA

Select[Range[2, 50000], OddQ[ # ] && ! PrimeQ[ # ] && Mod[Fibonacci[ # ] - 1, # ] == 0 &]

CROSSREFS

Cf. A094395, A094400.

Sequence in context: A083138 A121209 A065884 this_sequence A094409 A006465 A088214

Adjacent sequences: A094391 A094392 A094393 this_sequence A094395 A094396 A094397

KEYWORD

nonn

AUTHOR

Eric S Rowland (erowland(AT)math.rutgers.edu), May 01 2004

A094396 Numbers n such that n divides the (n-1)st Lucas number. +0
5
3, 4, 11476, 80476, 192676, 317683, 542242, 934876, 1339516, 4455676, 5063356, 7159636, 9004876, 9874684, 10134316, 17594242, 20558476 (list; graph; listen)
OFFSET

1,1
MATHEMATICA

Do[If[Mod[Fibonacci[n-2] + Fibonacci[n], n] == 0, Print[n]], {n, 3, 100000}] (Propper)

CROSSREFS

Cf. A094397.

Sequence in context: A069970 A153063 A165499 this_sequence A161838 A152624 A059107

Adjacent sequences: A094393 A094394 A094395 this_sequence A094397 A094398 A094399

KEYWORD

more,nonn

AUTHOR

Eric S Rowland (erowland(AT)math.rutgers.edu), May 01 2004

EXTENSIONS

10 more terms from Ryan Propper (rpropper(AT)stanford.edu), Jun 21 2005

More terms from Ryan Propper (rpropper(AT)cs.stanford.edu), Jan 03 2008

A094400 Odd n dividing Fibonacci(n)-1 but neither Fibonacci(n-1) nor Fibonacci(n+1). +0
5
7743, 27071, 54839, 72831, 217257, 388367, 417601, 575599, 670879, 691447, 701569, 809999, 850541, 881011 (list; graph; listen)
OFFSET

0,1
MATHEMATICA

Select[Range[50000], OddQ[ # ] && Mod[Fibonacci[ # ] - 1, # ] == 0 && ! Mod[Fibonacci[ # - 1], # ] == 0 && ! Mod[Fibonacci[ # + 1], # ] == 0 &]

CROSSREFS

Cf. A069106, A069107, A094394, A094401, A094402.

Sequence in context: A092004 A146960 A116239 this_sequence A116282 A052050 A116064

Adjacent sequences: A094397 A094398 A094399 this_sequence A094401 A094402 A094403

KEYWORD

nonn

AUTHOR

Eric S Rowland (erowland(AT)math.rutgers.edu), May 01 2004

A094603 a(n) is the length of the maximal sequence of rightmost black cells in the n-th row of Rule 30 (begun from an initial black cell). +0
5
1, 3, 1, 4, 1, 3, 1, 6, 1, 3, 1, 4, 1, 3, 1, 7, 1, 3, 1, 4, 1, 3, 1, 6, 1, 3, 1, 4, 1, 3, 1, 9, 1, 3, 1, 4, 1, 3, 1, 6, 1, 3, 1, 4, 1, 3, 1, 7, 1, 3, 1, 4, 1, 3, 1, 6, 1, 3, 1, 4, 1, 3, 1, 15, 1, 3, 1, 4, 1, 3, 1, 6, 1, 3, 1, 4, 1, 3, 1, 7, 1, 3, 1, 4, 1, 3, 1, 6, 1, 3, 1, 4, 1, 3, 1, 9, 1, 3, 1, 4, 1, 3, 1, 6, 1 (list; graph; listen)
OFFSET

1,2
COMMENT

New numbers (A094604) in the sequence occur at positions 2^m.

REFERENCES

Wolfram, Stephen. A New Kind of Science. Wolfram Media, 2002.

LINKS

Eric Weisstein's World of Mathematics, Rule 30

CROSSREFS

Cf. A007814, A094604.

Sequence in context: A004592 A116992 A090740 this_sequence A165595 A143825 A010602

Adjacent sequences: A094600 A094601 A094602 this_sequence A094604 A094605 A094606

KEYWORD

easy,nice,nonn

AUTHOR

Eric S Rowland (erowland(AT)math.rutgers.edu), May 13 2004

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