A055642 -id:A055642 - cp's OEIS frontend

A101337 Sum of (each digit of n raised to the power (number of digits in n)).

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 5, 10, 17, 26, 37, 50, 65, 82, 4, 5, 8, 13, 20, 29, 40, 53, 68, 85, 9, 10, 13, 18, 25, 34, 45, 58, 73, 90, 16, 17, 20, 25, 32, 41, 52, 65, 80, 97, 25, 26, 29, 34, 41, 50, 61, 74, 89, 106, 36, 37, 40, 45, 52, 61, 72, 85, 100, 117, 49, 50, 53, 58, 65

Offset: 1

Gordon Hamilton, Dec 24 2004

Sometimes referred to as "narcissistic function" (in base 10). Fixed points are the narcissistic (or Armstrong, or plus perfect) numbers A005188. - M. F. Hasler, Nov 17 2019

			a(75) = 7^2 + 5^2 = 74 and a(705) = 7^3 + 0^3 + 5^3 = 468.
a(1.02e59 - 1) = 102429587095122578993551250282047487264694110769657513064859 ~ 1.024e59 is an example of n close to the limit beyond which a(n) < n for all n. - _M. F. Hasler_, Nov 17 2019

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Wikipedia, Narcissistic number, as of Nov 18 2019.

Cf. A055642, A179239, A306360.

f:=func; [f(n):n in [1..75]]; // Marius A. Burtea, Nov 18 2019

Array[Total[IntegerDigits[#]^IntegerLength[#]]&,80] (* Harvey P. Dale, Aug 27 2011 *)

a(n)=my(d=digits(n)); sum(i=1,#d, d[i]^#d) \\ Charles R Greathouse IV, Aug 10 2017

apply( A101337(n)=vecsum([d^#n|d<-n=digits(n)]), [0..99]) \\ M. F. Hasler, Nov 17 2019

def A101337(n):
    s = str(n)
    l = len(s)
    return sum(int(d)**l for d in s) # Chai Wah Wu, Feb 26 2019

a(n) <= A055642(n)*9^A055642(n) with equality for all n = 10^k - 1. Write n = 10^x to get a(n) < n when 1+log_10(x+1) < (x+1)(1-log_10(9)) <=> x > 59.85. It appears that a(n) < n already for all n > 1.02*10^59. - M. F. Hasler, Nov 17 2019

Name changed by Axel Harvey, Dec 26 2011

Edited by M. F. Hasler, Nov 17 2019

A235224 a(0) = 0, and for n > 0, a(n) = largest k such that A002110(k-1) <= n, where A002110(k) gives the k-th primorial number.

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

Reinhard Zumkeller, Jan 05 2014

nonn

For n > 0: a(n) = (length of row n in A235168) = A055642(A049345(n)).

For n > 0, a(n) gives the length of primorial base expansion of n. Also, after zero, each value n occurs A061720(n-1) times. - Antti Karttunen, Oct 19 2019

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to primorial base

Cf. A000040, A002110, A049345, A055642, A061395, A061720, A084558, A267263, A276086, A235168, A328114, A328404, A328405, A328406.

a235224 n = length $ takeWhile (<= n) a002110_list

A235224 := proc(n)
    local k;
    if n = 0 then
        0;
    else
        for k from 0 do
            if A002110(k-1) > n then
                return k-1 ;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Apr 19 2021

primorial[n_] := Times @@ Prime[Range[n]];
a[n_] := TakeWhile[primorial /@ Range[0, n], # <= n &] // Length;
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 27 2021 *)

A235224(n) = { my(s=0, p=2); while(n, s++; n = n\p; p = nextprime(1+p)); (s); }; \\ Antti Karttunen, Oct 19 2019

A235224(n, p=2) = if(!n,n,if(nA235224(n\p, nextprime(p+1)))); \\ (Recursive implementation) - Antti Karttunen, Oct 19 2019

From Antti Karttunen, Oct 19 2019: (Start)
a(n) = A061395(A276086(n)).
For all n >= 0, a(n) >= A267263(n).
For all n >= 1, A000040(a(n)) > A328114(n). (End)

Name corrected to match the data by Antti Karttunen, Oct 19 2019

A028335 Number of decimal digits in n-th Mersenne prime.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 6, 10, 19, 27, 33, 39, 157, 183, 386, 664, 687, 969, 1281, 1332, 2917, 2993, 3376, 6002, 6533, 6987, 13395, 25962, 33265, 39751, 65050, 227832, 258716, 378632, 420921, 895932, 909526, 2098960, 4053946, 6320430, 7235733, 7816230, 9152052, 9808358, 11185272

Offset: 1

N. J. A. Sloane

			A000668(6) = 2^17-1 = 131071 has 6 decimal digits, so a(6) = 6.
A000668(10) = 2^89-1 = 618,970,019,642,690,137,449,562,111 has 27 digits, so a(10) = 27.

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 19.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..48 (terms 1..47 from Ivan Panchenko)
Chris K. Caldwell, Mersenne Primes.
GIMPS, List of Known Mersenne Prime Numbers.
Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2023.

See A000043, which is the main entry for this sequence.

Cf. A000668, A055642.

seq(length(numtheory:-mersenne([i])),i=1..45); # Robert Israel, Feb 02 2018

IntegerLength[2^Array[MersennePrimeExponent, 45] - 1] (* Jean-François Alcover, Feb 17 2018 *)
a[n_] := Floor[MersennePrimeExponent[n]/Log2[10]] + 1; Array[a, 48] (* Amiram Eldar, Oct 16 2024 *)

a(n) = floor(A000043(n)*log(2)/log(10)) + 1.

a(n) = A055642(A000668(n)). - Michel Marcus, Apr 07 2018

More terms from Enoch Haga, Dec 18 2001
a(38) from Harry J. Smith, Apr 17 2003
a(39) from Omar E. Pol, Oct 28 2007
a(40)-a(41) from Jason Kimberley, Jan 05 2012
a(42)-a(45) from Patrick J. McNab, Feb 01 2018

A178788 Characteristic function of numbers having distinct digits in their decimal representation.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1

Offset: 0

Reinhard Zumkeller, Jun 30 2010

a(A010784(n)) = 1; a(A109303(n)) = 0;
first differences of A178787.
a(n) <= A196368(n).
a(n) = 0 for n > 9*9!. - Hieronymus Fischer, Feb 02 2013

R. Zumkeller, Table of n, a(n) for n = 0..10000
Michael De Vlieger, Plot a(n) at (x,y) = (n mod 1000, -floor(n/1000)), n = 1..10^6, where black represents a(n) = 1 and white represents a(n) = 0.
Eric Weisstein's World of Mathematics, Digit
Index entries for characteristic functions

import Data.List (nub)
a178788 n = fromEnum $ nub (show n) == show n
-- Reinhard Zumkeller, Sep 25 2011

lst = {}; Do[If[Select[Tally[IntegerDigits[n]][[All, 2]], # > 1 &] == {}, AppendTo[lst, 1], AppendTo[lst, 0]], {n, 0, 104}]; lst (* Arkadiusz Wesolowski, May 10 2012 *)

a(n) = 0^(A055642(n)-A043537(n)).

A060384 Number of decimal digits in n-th Fibonacci number.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18

Offset: 0

Labos Elemer, Apr 03 2001

Harry J. Smith, Table of n, a(n) for n = 0..10000

Cf. A000045, A022307, A001605, A060319, A060320, A051694, A050815.

Cf. A261585.

a060384 = a055642 . a000045  -- Reinhard Zumkeller, Mar 09 2013

with(combinat): a:=n->nops(convert(fibonacci(n),base,10)): 1,seq(a(n),n=1..100); # Emeric Deutsch, May 19 2007

Table[IntegerLength@ Fibonacci@ n, {n, 0, 84}] /. 0 -> 1 (* or *)
Table[Floor[n Log10@ GoldenRatio - Log10@ 5/2] + 1, {n, 0, 84}] /. 0 -> 1 (* Michael De Vlieger, Jul 04 2016 *)

print1("1, 1, "); gold=(1+sqrt(5))/2; for(n=2,100,print1(floor((n*log(gold)-log(5)/2)/log(10))+1", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007

a(n) = #Str(fibonacci(n)); \\ Michel Marcus, Jul 04 2016

a(n) = floor(n*log(tau)/log(10)) +0 or +1 where tau is the golden ratio. - Benoit Cloitre, Oct 29 2002. [Corrected by Hans J. H. Tuenter, Jul 07 2025].
a(n) = floor(n*log_10(gold) - log_10(5)/2) + 1 for n >= 2, where gold is (1+sqrt(5))/2. - Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007
a(n) = A055642(A000045(n)). - Reinhard Zumkeller, Mar 09 2013

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007

A067488 Powers of 2 with initial digit 1.

Original entry on oeis.org

1, 16, 128, 1024, 16384, 131072, 1048576, 16777216, 134217728, 1073741824, 17179869184, 137438953472, 1099511627776, 17592186044416, 140737488355328, 1125899906842624, 18014398509481984, 144115188075855872, 1152921504606846976, 18446744073709551616

Offset: 1

Amarnath Murthy, Feb 09 2002

Also smallest n-digit power of 2.

For each range 10^(n-1) to 10^n-1 there exists exactly 1 power of 2 with first digit 1 (floor(log_10(a(n))) = n-1). As such, the density of this sequence relative to all powers of 2 (A000079) is log(2)/log(10) (0.301..., A007524), which is prototypical of Benford's Law. - Charles L. Hohn, Jul 23 2024

Muniru A Asiru, Table of n, a(n) for n = 1..993
Index to divisibility sequences

Cf. A000079, A067497, A055642,
Other initial digits: A067480, A067481, A067482, A067483, A067484, A067485, A067486, A067487, A074116.
Cf. A074117, A074118.

Filtered(List([0..60],n->2^n),i->ListOfDigits(i)[1]=1); # Muniru A Asiru, Oct 22 2018

[2^n: n in [0..100] | Intseq(2^n)[#Intseq(2^n)] eq 1]; // Vincenzo Librandi, Dec 31 2024

Select[2^Range[0, 70], First[IntegerDigits[#]] == 1 &]  (* Harvey P. Dale, Mar 14 2011 *)

a(n)=2^ceil((n-1)*log(10)/log(2)) \\ Charles R Greathouse IV, Apr 08 2012

(List.fill(50)(2: BigInt)).scanLeft(1: BigInt)( * ).filter(.toString.startsWith("1")) // _Alonso del Arte, Jan 16 2020

a(n) = 2^ceiling((n-1)*log(10)/log(2)). - Benoit Cloitre, Aug 29 2002
From Charles L. Hohn, Jun 09 2024: (Start)
a(n) = 2^A067497(n-1).
A055642(a(n)) = n. (End)

A045876 Sum of different permutations of digits of n (leading 0's allowed).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 11, 33, 44, 55, 66, 77, 88, 99, 110, 22, 33, 22, 55, 66, 77, 88, 99, 110, 121, 33, 44, 55, 33, 77, 88, 99, 110, 121, 132, 44, 55, 66, 77, 44, 99, 110, 121, 132, 143, 55, 66, 77, 88, 99, 55, 121, 132, 143, 154, 66, 77, 88, 99, 110, 121, 66, 143

Offset: 1

Erich Friedman

Let the arithmetic mean of the digits of a 'D' digit number n be 'A', let 'N' = number of distinct numbers that can be formed by permuting the digits of n, and let 'I' = concatenation of 1 'D' times = (10^D-1)/9. then a(n) = A*N*I. E.g., let n = 324541, then A = (3+2+4+5+4+1)/6 = 19/6, N = 6!/(2!) = 360, I = 111111, and a(n) = A*N*I = (19/6)*(360)*(111111) = 126666540. - Amarnath Murthy, Jul 14 2003

It seems that the first person who studied the sum of different permutations of digits of a given number was the French scientist Eugène Aristide Marre (1823-1918). See links. - Bernard Schott, Dec 06 2012

Amarnath Murthy, An interesting result in combinatorics, Mathematics & Informatics Quarterly, Vol. 3, 1999, Bulgaria.

A. Dunigan AtLee, Table of n, a(n) for n = 1..100000.
A. Marre, Trouver la somme de toutes les permutations différentes d'un nombre donné., Nouvelles Annales de Mathématiques, 1ère série, tome 5 (1846), p. 57-60.
Norbert Verdier, QDV4 : Marre, Marre et Marre, page=1 (French mathematical forum les-mathematiques.net)

Same beginning as A033865. Cf. A061147.

f:= proc(x) local L,D,n,M,s,j;
  L:= convert(x,base,10);
  D:= [seq(numboccur(j,L),j=0..9)];
  n:= nops(L);
  M:= n!/mul(d!,d=D);
  s:= add(j*D[j+1],j=0..9);
  (10^n-1)*M/9/n*s
end proc:
map(f, [$1..100]); # Robert Israel, Jul 07 2015

Table[Total[FromDigits /@ Permutations[IntegerDigits[n]]], {n, 100}] (* T. D. Noe, Dec 06 2012 *)

A047726(n) = n=eval(Vec(Str(n))); (#n)!/prod(i=0, 9, sum(j=1, #n, n[j]==i)!);
A055642(n) = #Str(n);
A007953(n) = sumdigits(n);
a(n) = ((10^A055642(n)-1)/9)*(A047726(n)*A007953(n)/A055642(n)); \\ Altug Alkan, Aug 29 2016

A045876(n) = {my(d=digits(n), q=1, v, t=1); v = vecsort(d); for(i=1, #v-1, if(v[i]==v[i+1], t++, q*=binomial(i, t); t=1)); q*binomial(#v, t)*(10^#d-1)*vecsum(d)/9/#d} \\ David A. Corneth, Oct 06 2016

a(n) = ((10^A055642(n)-1)/9)*(A047726(n)*A007953(n)/A055642(n)). - Altug Alkan, Aug 29 2016

A046759 Economical numbers: write n as a product of primes raised to powers, let D(n) = number of digits in product, l(n) = number of digits in n; sequence gives n such that D(n) < l(n).

Original entry on oeis.org

125, 128, 243, 256, 343, 512, 625, 729, 1024, 1029, 1215, 1250, 1280, 1331, 1369, 1458, 1536, 1681, 1701, 1715, 1792, 1849, 1875, 2048, 2187, 2197, 2209, 2401, 2560, 2809, 3125, 3481, 3584, 3645, 3721, 4096, 4374, 4375, 4489, 4802, 4913

Offset: 1

N. J. A. Sloane, Dec 11 1999

A050252(a(n)) < A055642(a(n)). - Reinhard Zumkeller, Jun 21 2011

			125 = 5^3, l(n) = 3 and D(n) = 2, so 125 is a member of the sequence.

Bernardo Recamán, The Bogota Puzzles, Courier Dover Publications, Inc., 2020, p. 77.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
C. K. Caldwell, The Prime Glossary, economical number.
J.-M. De Koninck and F. Luca, On strings of consecutive economical numbers of arbitrary length, INTEGERS (2005), Volume: 5, Issue: 2, #A5.
Jean-Paul Delahaye, Les chasseurs de nombres premiers, Pour la Science, No. 258 April 1999. [v1].
R. G. E. Pinch, Economical numbers.
R. G. E. Pinch, Economical numbers, arXiv:math/9802046 [math.NT], 1998.
W. Schneider, Economical Numbers.
G. Villemin's Almanach of Numbers, Nombres Economes.
Eric Weisstein's World of Mathematics, Economical Number.
Wikipedia, Frugal number.

Cf. A046758, A046760.

a046759 n = a046759_list !! (n-1)
a046759_list = filter (\n -> a050252 n < a055642 n) [1..]
-- Reinhard Zumkeller, Jun 21 2011

ecoQ[n_] := Total[ Length /@ IntegerDigits /@ Flatten[ FactorInteger[n] /. {p_, 1} -> p]] < Length[ IntegerDigits[n]]; Select[ Range[5000], ecoQ] (* Jean-François Alcover, Jul 28 2011 *)

is(n)=my(f=factor(n));sum(i=1,#f[,1], #Str(f[i,1])+if(f[i,2]>1, #Str(f[i,2])))<#Str(n) && n>1 \\ Charles R Greathouse IV, Feb 01 2013

from sympy import factorint
def ok(n): return n > 1 and sum(len(str(p))+(len(str(e)) if e>1 else 0) for p, e in factorint(n).items()) < len(str(n))
print([k for k in range(5000) if ok(k)]) # Michael S. Branicky, Dec 22 2024

More terms from Eric W. Weisstein

A292730 Numbers in which 0 outnumbers all other digits together.

Original entry on oeis.org

0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000, 10000, 10001, 10002, 10003, 10004, 10005, 10006, 10007, 10008, 10009, 10010, 10020, 10030, 10040, 10050, 10060, 10070, 10080, 10090, 10100, 10200, 10300, 10400, 10500, 10600, 10700, 10800, 10900, 11000

Offset: 1

Halfdan Skjerning, Sep 22 2017

Subset of A292450.
Numbers n such that A055641(n) > (A055642(n)/2). - Felix Fröhlich, Sep 22 2017
Also numbers whose median of the digits is equal to 0. - Stefano Spezia, Oct 04 2023

			100 has more 0's than any other digit, whereas both 1001 and 1002 have as many other digits as 0's.

Cf. A292449, A292450, A292451, A292452, A292453, A292454, A292455, A292456, A292457, A292458, A292731, A292732, A292733, A292734, A292735, A292736, A292737, A292738, A292739.

Cf. A055641, A055642.

Select[Range[0, 11000], Total@ #1 < First@ #2 & @@ TakeDrop[DigitCount@ #, 9] &] (* Michael De Vlieger, Sep 22 2017 *)

a055641(n)=if(n, n=digits(n); sum(i=2, #n, n[i]==0), 1) \\ after Charles R Greathouse IV
is(n) = a055641(n) > (#Str(n)/2) \\ Felix Fröhlich, Sep 22 2017

A054382 James Joyce's "Ulysses" sequence: number of digits in n^(n^n).

Original entry on oeis.org

1, 1, 2, 13, 155, 2185, 36306, 695975, 15151336, 369693100, 10000000001, 297121486765, 9622088391635, 337385711567665, 12735782555419983, 515003176870815368, 22212093154093428530, 1017876887958723919835, 49390464231494436119285

Offset: 0

Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), May 07 2000

Although Joyce mentions (9^9)^9, he clearly intended to refer to 9^(9^9).

(9^9)^9 is only 196627050475552913618075908526912116283103450944214766927315415537966391196809, whereas 9^(9^9) has 369693100 digits.

			"Because some years previously in 1886 when occupied with the problem of the quadrature of the circle he had learned of the existence of a number computed to a relative degree of accuracy to be of such magnitude and of so many places, e.g., the 9th power of the 9th power of 9, that, the result having been obtained, 33 closely printed volumes of 1000 pages each of innumerable quires and reams of India paper would have to be requisitioned in order to contain the complete tale of its printed integers of units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions, hundreds of millions, billions, the nucleus of the nebula of every digit of every series containing succinctly the potentiality of being raised to the utmost kinetic elaboration of any power of any of its powers." - James Joyce, Ulysses, Chapter 17.
a(2)=2 since 2^(2^2)=2^4=16 has 2 digits. - _Carmine Suriano_, Feb 01 2011
a(0)=1 because 0^(0^0)=0^1=0, which has 1 digit. - _T. D. Noe_, Feb 01 2011

C. A. Laisant (1906) proved that the number of digits of a(9), 9^9^9, is 369693100. H. S. Uhler (1947) published the log of the number to 250 decimal places.
David Wells: The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, 1986, p. 208.

Robert G. Wilson v, Table of n, a(n) for n = 0..1000 (first 57 terms from Carmine Suriano)
H. Havermann, 9^9^9, in 33 volumes.
J. Joyce, Ulysses, Ithaca chapter
Eric Weisstein's World of Mathematics, Joyce Sequence.
Eric Weisstein's World of Mathematics, Sultan's Dowry Problem.

Cf. A002488, A055642, A066022.

A055642 := proc(n) max(1,ilog10(n)+1) ; end proc:
A054382 := proc(n) A055642(n^(n^n)) ; end proc: # R. J. Mathar, Feb 01 2011

f[ j_ ] := 1 + Floor[ Log[10, j] j^j ]; Table[ f[j], {j, 2, 20} ]

a(n) = floor(n^n*log_10(n)) + 1 for n > 0. - Jianing Song, Nov 21 2018

More terms from Michael Kleber, May 07 2000

cp's OEIS Frontend

A101337 Sum of (each digit of n raised to the power (number of digits in n)).

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A235224 a(0) = 0, and for n > 0, a(n) = largest k such that A002110(k-1) <= n, where A002110(k) gives the k-th primorial number.

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A028335 Number of decimal digits in n-th Mersenne prime.

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A178788 Characteristic function of numbers having distinct digits in their decimal representation.

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A060384 Number of decimal digits in n-th Fibonacci number.

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A067488 Powers of 2 with initial digit 1.

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