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Smarandache Number


Consider the Consecutive Number Sequences formed by the concatenation of the first n positive integers: 1, 12, 123, 1234, ... (OEIS A007908; Smarandache 1993, Dumitrescu and Seleacu 1994, sequence 1; Mudge 1995; Stephan 1998; Wolfram 2002, p. 913). This sequence gives the digits of the Champernowne constant, and is sometimes also known as the Barbier infinite word (Allouche and Shallit 2003, pp. 114, 299, and 336). The terms up to n=9 are given by

c_n=sum_(k=1)^(n)k·10^(n-k)
(1)
=1/(81)(10^(n+1)-9n-10).
(2)

These are sometimes called Smarandache consecutive numbers, but in this work, the terms in the sequence will be called simply Smarandache numbers. Similarly, a Smarandache number that is prime will be called a Smarandache prime. Surprisingly, no Smarandache primes Sm(n) exist for n<=344869 (Great Smarandache PRPrime search; Dec. 5, 2016).

The number of digits of Sm(n) can be computed by noticing the pattern in the following table, where

 d=|_log_(10)n_|+1
(3)

is the number of digits in n.

dn rangedigits
11-9n
210-999+2(n-9)
3100-9999+90·2+3(n-99)
41000-99999+90·2+900·3+4(n-999)

By induction, the number of digits D(n) in Sm(n) can be written

D(n)=d(n+1-10^(d-1))+sum_(k=1)^(d-1)9k·10^(k-1)
(4)
=(n+1)d-(10^d-1)/9,
(5)

where the second term is the repunit R_d. For n=1, 2, ..., the digit lengths D(n) of Sm(n) are therefore 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, ... (OEIS A058183).

Plots of the concatenation of consecutive integers in base 2

The results of concatenating the binary representations of the first few integers are 1, 110, 11011, 11011100, 11011100101, ... (OEIS A058935). These digit sequences are plotted above for n=1 to 90. Interpreting the digit sequence as a binary fraction, the result is the binary Champernowne constant C_2.

ConsecutiveIntegersCumulativeSum

Interestingly, taking the cumulative sum 2x_i-1 where {x_i} are the digits C_2 gives a plot showing batrachion-like structure (left figure), and doing the same with {x_i}_(i=2)^infty (right figure) gives structures resembling the Blancmange function (and the Hofstadter-Conway $10,000 sequence).


See also

Champernowne Constant, Champernowne Constant Digits, Consecutive Number Sequences, Integer Sequence Primes, Smarandache Prime, Smarandache Sequences

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References

--. "The Great Smarandache PRPrime search." http://smarandache.ddns.net:1200/server_stats.html.Mudge, M. "Top of the Class." Personal Computer World, 674-675, June 1995.Mudge, M. "Not Numerology but Numeralogy!" Personal Computer World, 279-280, 1997.Sloane, N. J. A. Sequences A007908, A058183, and A058935 in "The On-Line Encyclopedia of Integer Sequences."Smarandache, F. Only Problems, Not Solutions!, 4th ed. Phoenix, AZ: Xiquan, 1993.Stephan, R. W. "Factors and Primes in Two Smarandache Sequences." Smarandache Notions J. 9, 4-10, 1998.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 913, 2002.

Cite this as:

Weisstein, Eric W. "Smarandache Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SmarandacheNumber.html

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