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A007924
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The number n written using the greedy algorithm in the base where the values of the places are 1 and primes.
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8
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0, 1, 10, 100, 101, 1000, 1001, 10000, 10001, 10010, 10100, 100000, 100001, 1000000, 1000001, 1000010, 1000100, 10000000, 10000001, 100000000, 100000001, 100000010, 100000100, 1000000000, 1000000001, 1000000010, 1000000100, 1000000101
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OFFSET
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0,3
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COMMENTS
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Any nonnegative number can be written as a sum of distinct primes + e, where e is 0 or 1.
Terms contain only digits 0 and 1.
Without the "greedy" condition there is ambiguity - for example 5 = 3+2 has two representations.
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REFERENCES
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Florian Luca & Ravindranathan Thangadurai, "On an arithmetic function considered by Pillai", Journal de theorie des nombres de Bordeaux 21:3 (2009), pp. 695-701.
S. S. Pillai, "An arithmetical function concerning primes", Annamalai University Journal (1930), pp. 159-167.
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LINKS
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Table of n, a(n) for n=0..27.
C. Rivera, Prime puzzle 78
F. Smarandache, Only Problems, Not Solutions!
F. Smarandache, Definitions, Solved and Unsolved Problems, Conjectures, and Theorems in Number Theory and Geometry, edited by M. Perez, Xiquan Publishing House 2000.
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FORMULA
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a(n) is the binary representation of b(n) = 2^pi(n) + b(n-p(pi(n))) for n > 0 and a(0) = b(0)= 0, where pi(k) = number of primes <= k (A000720) and p(k) = k-th prime (A008578). - Frank Ellermann, Dec 18, 2001
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EXAMPLE
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4 = 3 + 1, so a(4) = 101.
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MATHEMATICA
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cprime[n_Integer] := (If[n==0, 1, Prime[n]]); gentable[n_Integer] := (m=n; ptable={}; While[m!=0, (i=0; While[cprime[i]<=m, i++]; j=0; While[j<i, AppendTo[ptable, 0]; j++]; ptable[[i]]=1; m=m-cprime[i-1])]; ptable); decimal[n_Integer] := (gentable[n]; Sum[2^(k-1)*ptable[[k]], {k, 1, Length[ptable]}]); Table[IntegerString[decimal[n], 2], {n, 0, 100}](* Frank M. Jackson, Jan 06 2012 *)
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PROG
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(PARI) a(n)=if(n>1, my(p=precprime(n)); 10^primepi(p)+a(n-p), n) \\ Charles R Greathouse IV, Feb 01 2013
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CROSSREFS
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Cf. A200947, A066352.
Sequence in context: A014417 A211027 A185101 * A115794 A105424 A115832
Adjacent sequences: A007921 A007922 A007923 * A007925 A007926 A007927
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KEYWORD
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nonn,easy
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AUTHOR
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R. Muller
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EXTENSIONS
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Additional references from Felice Russo, Sep 14 2001
Antedated to 1930 by Charles R Greathouse IV, Aug 28 2010
Definition clarified by Frank M Jackson and N. J. A. Sloane, Dec 30 2011
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STATUS
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approved
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