OFFSET
0,3
COMMENTS
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.
REFERENCES
S. S. Pillai, "An arithmetical function concerning primes", Annamalai University Journal (1930), pp. 159-167.
LINKS
John Cerkan, Table of n, a(n) for n = 0..5000
K. Kashihara, Comments and Topics on Smarandache Notions and Problems, Erhus University Press, 1996, 50 pages. See page 33.
K. Kashihara, Comments and Topics on Smarandache Notions and Problems, Erhus University Press, 1996, 50 pages. [Cached copy] See page 33.
Florian Luca & Ravindranathan Thangadurai, On an arithmetic function considered by Pillai, Journal de théorie des nombres de Bordeaux 21:3 (2009), pp. 695-701.
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.
FORMULA
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
EXAMPLE
4 = 3 + 1, so a(4) = 101.
MATHEMATICA
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 *)
PROG
(PARI) a(n)=if(n>1, my(p=precprime(n)); 10^primepi(p)+a(n-p), n) \\ Charles R Greathouse IV, Feb 01 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. Muller
EXTENSIONS
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
STATUS
approved