A038399 -id:A038399 - cp's OEIS frontend

A134069 Primes in A038399.

11, 211, 853211

Offset: 1

Alexander Adamchuk, Oct 06 2007

A038399 = {1, 11, 211, 3211, 53211, 853211, 13853211, 2113853211, ...} = concatenation of first n nonzero Fibonacci numbers in reverse order.
From Robert Israel, Sep 14 2016: (Start)
No more terms < A038399(500), which has 26252 digits.
The probability of a number of the order of magnitude of A038399(n) being prime is approximately constant/n^2. Since Sum_n 1/n^2 converges, we should expect this sequence to be finite. (End)

Eric Weisstein's World of Mathematics, Fibonacci Number
Eric Weisstein's World of Mathematics, Smarandache Sequences

Cf. A038399 (concatenation of first n nonzero Fibonacci numbers in reverse order).

Cf. A019523 (concatenation of Fibonacci numbers).

count:= 0:
A038399[1]:= 1:
for n from 2 to 100 do
A038399[n]:= combinat:-fibonacci(n)*10^(1+ilog10(A038399[n-1]))+A038399[n-1];
   if isprime(A038399[n]) then count:= count+1; A[count]:= A038399[n] fi
od:
seq(A[i],i=1..count); # Robert Israel, Sep 14 2016

Module[{nn=10,bif},bif=Fibonacci[Range[nn]];Select[Table[FromDigits[ Flatten[ IntegerDigits/@Reverse[Take[bif,n]]]],{n,nn}],PrimeQ]] (* Harvey P. Dale, Sep 27 2019 *)

A371720 a(n) = m^^m mod 10^len(m), where m = A038399(n) and ^^ indicates tetration or hyper-4.

Original entry on oeis.org

1, 11, 811, 3811, 63811, 763811, 3763811, 5103763811, 515103763811, 19515103763811, 6819515103763811, 8146819515103763811, 3808146819515103763811, 7213808146819515103763811, 9807213808146819515103763811, 4939807213808146819515103763811

Offset: 1

Marco Ripà, Apr 04 2024

For any n, a(n) == a(n + 1) (mod 10^len(A038399(n))), where len(k) := number of digits in k. Assuming len(a(n)) > 1, this is a general property of every concatenated sequence with fixed rightmost digits (such as A014925, A061839, A092845, and A104759), as shown in Ripà's book "La strana coda della serie n^n^...^n".

Moreover, assuming n > 1, since A038399(n) is congruent to 11 (mod 20), the convergence speed of A038399(n)^^b (say, V(A038399(n), b) = {2, 1, 1, 1, ...}) is 2 at height 1 and becomes a unit value for any integer b > 1 (see Links). Hence, a(n) is given by A038399(n)^^len(A038399(n) - 1) (mod 10^len(A038399(n))), and also by A038399(n)^^len(A038399(n)) (mod 10^len(A038399(n))) since A038399(n)^^len(A038399(n)) == A038399(n)^^len(A038399(n) - 1) (mod 10^len(A038399(n))) holds for any n.

			a(8) is given by the rightmost 10 digits of 2113853211^^2113853211 and thus a(8) = 5103763811.
a(9) == a(8) (mod 10^10), i.e., the digits of a(9) end with the digits of a(8) (and then a(9) has 2 more preceding).

Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011, page 60. ISBN 978-88-6178-789-6

Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43-61.
Marco Ripà and Luca Onnis, Number of stable digits of any integer tetration, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441-457.
Wikipedia, Tetration.

Cf. A000045 (Fibonacci), A038399, A171882 (tetration), A317824, A317903, A317905.

a(n) = A038399(n)^^(len(A038399(n)) - 1) mod 10^len(A038399(n)), where len(A038399(n)) = ceiling(log_10(A038399(n) + 1)).

A019523 Concatenation of Fibonacci(1) through Fibonacci(n).

Original entry on oeis.org

1, 11, 112, 1123, 11235, 112358, 11235813, 1123581321, 112358132134, 11235813213455, 1123581321345589, 1123581321345589144, 1123581321345589144233, 1123581321345589144233377, 1123581321345589144233377610, 1123581321345589144233377610987

Offset: 1

R. Muller

For n<=800, only a(2) and a(4) are primes. - Dmitry Kamenetsky, Feb 25 2009

a(n) has about kn(n+1) digits, where k = log phi/log 100 = 0.10449... - Charles R Greathouse IV, Sep 19 2012

S. Smarandoiu, Convergence of Smarandache continued fractions, Abstract 96T-11-195, Abstracts Amer. Math. Soc., 17 (No. 4, 1996), 680.

Reinhard Zumkeller, Table of n, a(n) for n = 1..96
F. Smarandache, Collected Papers, Vol. II
Eric Weisstein's World of Mathematics, Fibonacci Number
Eric Weisstein's World of Mathematics, Consecutive Number Sequences
Eric Weisstein's World of Mathematics, Smarandache Sequences
Index entries for sequences related to Most Wanted Primes video

Cf. A000045, A038399.

a019523 n = read $ concatMap show $ take n $ tail a000045_list :: Integer
-- Reinhard Zumkeller, Mar 01 2014

[Seqint(Reverse(&cat[Reverse(Intseq(Fibonacci(k))): k in [1..n]])): n in [1..20]]; // Vincenzo Librandi, Dec 18 2016

Table[FromDigits[Flatten[IntegerDigits[Fibonacci[Range[n]]]]], {n,25}] (* G. C. Greubel, Nov 30 2016 *)

a(n) = a(n-1)*10^A055642(A000045(n)) + A000045(n). - José de Jesús Camacho Medina, Dec 16 2016

A038395 Concatenation of the first n odd numbers in reverse order.

Original entry on oeis.org

1, 31, 531, 7531, 97531, 1197531, 131197531, 15131197531, 1715131197531, 191715131197531, 21191715131197531, 2321191715131197531, 252321191715131197531, 27252321191715131197531, 2927252321191715131197531, 312927252321191715131197531

Offset: 1

M. I. Petrescu (mipetrescu(AT)yahoo.com)

a(n) starts with the digits of 2n-1. Indices of prime or probable prime terms are 1,2,37,62,409,...: see also A089922. - M. F. Hasler, Apr 13 2008

If n == 0 (mod 3), so is a(n). - Sergey Pavlov, Mar 29 2017

Mihaly Bencze [Beneze] and L. Tutescu, Some Notions and Questions in Number Theory, Sequence 3.

Florentin Smarandache, Sequences of Numbers Involved in Unsolved Problems, arXiv:math/0604019 [math.GM], 2006.

Cf. A089922, A109837, A038394-A038399, A019518, A019519.

Table[FromDigits[Flatten[IntegerDigits/@Join[Reverse[Range[1,n,2]]]]], {n,1,29,2}] (* Harvey P. Dale, Jun 02 2011 *)

t=""; for( n=1,10^3, ( t=eval( Str( 2*n-1,t))) & print(n" "t)) \\ M. F. Hasler, Apr 13 2008

def a(n): return int("".join(map(str, range(2*n-1, 0, -2))))
print([a(n) for n in range(1, 17)]) # Michael S. Branicky, Jan 31 2021

Edited and extended by M. F. Hasler, Apr 13 2008

Edited by T. D. Noe, Oct 30 2008

A134072 Concatenation of A000204 Lucas numbers (beginning at 1) in reverse order.

Original entry on oeis.org

1, 31, 431, 7431, 117431, 18117431, 2918117431, 472918117431, 76472918117431, 12376472918117431, 19912376472918117431, 32219912376472918117431, 52132219912376472918117431, 84352132219912376472918117431

Offset: 1

Alexander Adamchuk, Oct 06 2007

Indices of prime terms are {2, 3, 5, 11, ...}. Primes are listed in A134071 = {31, 431, 117431, 19912376472918117431, ...}.

Eric Weisstein's World of Mathematics, Lucas Number
Eric Weisstein's World of Mathematics, Smarandache Sequences

Cf. A000204 (Lucas numbers).
Cf. A130774 (concatenation of Lucas numbers).
Cf. A019523 (concatenation of Fibonacci numbers).
Cf. A038399 (concatenation of first n nonzero Fibonacci numbers in reverse order).
Cf. A134069 (primes in A038399).
Cf. A134070 (primes in A130774).
Cf. A134071 (primes in A134072).

Module[{nn=20,lnos},lnos=LucasL[Range[nn]];Table[FromDigits[Flatten[ IntegerDigits/@ Reverse[Take[lnos,n]]]],{n,nn}]] (* Harvey P. Dale, Jul 27 2015 *)

Edited by Charles R Greathouse IV, Apr 26 2010

A134070 Primes in A130774.

Original entry on oeis.org

13, 13471118294776123, 134711182947761231993225218431364220735715778934915127244763960364079103682167761271443439204710647114985118604983010349

Offset: 1

Alexander Adamchuk, Oct 06 2007

A130774 = {1, 13, 134, 1347, 134711, 13471118, 1347111829, 134711182947, 13471118294776, 13471118294776123, ...} = Concatenate Lucas numbers. Indices of prime terms in A130774 are {2, 10, 31, ...}.

a(4) > A130774(500) = 1.347...*10^26425, if it exists. - Amiram Eldar, Jul 17 2025

			a(1) = A130774(2) = 13.
a(2) = A130774(10) = 13471118294776123.
a(3) = A130774(31) = 13471118294776123199322521843136422073571577893491512724476396036407910368216776127144343920471064711498511860498301034.

Eric Weisstein's World of Mathematics, Lucas Number.
Eric Weisstein's World of Mathematics, Smarandache Sequences.

Cf. A000204 (Lucas numbers).
Cf. A130774 (concatenation of Lucas numbers).
Cf. A019523 (concatenation of Fibonacci numbers).
Cf. A038399 (concatenation of first n nonzero Fibonacci numbers in reverse order).
Cf. A134072 (concatenation of A000204 Lucas numbers (beginning at 1) in reverse order).
Cf. A134069 (primes in A038399).
Cf. A134071 (primes in A134072).

Select[Module[{nn=40,ll},ll=LucasL[Range[nn]];Table[FromDigits[Flatten[IntegerDigits/@Take[ll,n]]],{n,nn}]],PrimeQ] (* Harvey P. Dale, May 07 2023 *)

Edited by Charles R Greathouse IV, Apr 24 2010

A134071 Primes in A134072.

Original entry on oeis.org

31, 431, 117431, 19912376472918117431

Offset: 1

Alexander Adamchuk, Oct 06 2007

A134072 = {1, 31, 431, 7431, 117431, 18117431, 2918117431, 472918117431, 76472918117431, 12376472918117431, 19912376472918117431, ...} = concatenation of Lucas numbers in reverse order. Indices of prime terms in A134072 are {2, 3, 5, 11, ...}.

No further terms through 500 Lucas numbers. - Harvey P. Dale, Aug 17 2013

			a(1) = A134072(2) = 31.
a(2) = A134072(3) = 431.
a(3) = A134072(5) = 117431.

Eric Weisstein's World of Mathematics, Lucas Number
Eric Weisstein's World of Mathematics, Smarandache Sequences

Cf. A000204 (Lucas numbers).
Cf. A130774 (concatenation of Lucas numbers).
Cf. A019523 (concatenation of Fibonacci numbers).
Cf. A038399 (concatenation of first n nonzero Fibonacci numbers in reverse order).
Cf. A134072 (concatenation of A000204 Lucas numbers (beginning at 1) in reverse order).
Cf. A134069 (primes in A038399).
Cf. A134070 (primes in A130774).

nn=500;With[{lucs=LucasL[Range[nn]]},Select[Table[FromDigits[ Flatten[ IntegerDigits/@ Reverse[ Take[lucs,n]]]],{n,nn}],PrimeQ]] (* Harvey P. Dale, Aug 17 2013 *)

Edited by Charles R Greathouse IV, Apr 24 2010

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A134069 Primes in A038399.

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A371720 a(n) = m^^m mod 10^len(m), where m = A038399(n) and ^^ indicates tetration or hyper-4.

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A019523 Concatenation of Fibonacci(1) through Fibonacci(n).

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A038395 Concatenation of the first n odd numbers in reverse order.

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A134072 Concatenation of A000204 Lucas numbers (beginning at 1) in reverse order.

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A134070 Primes in A130774.

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A134071 Primes in A134072.

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