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 A102032 a(n) is the smallest positive integer k such that, if kn is written in base 2, it requires exactly n ones. 2
 1, 3, 7, 15, 11, 21, 89, 255, 167, 307, 349, 1365, 1259, 6729, 6417, 65535, 28431, 29127, 54757, 209715, 274627, 750685, 706649, 5592405, 2663383, 9679163, 14913005, 186946121, 37025579, 353440017, 1175487521, 4294967295 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS This sequence can be considered for any base b. If one calculates the arithmetic mean of the sequence d(n):=a(n)/2^n, i.e. (d(1)+d(2)+...+d(n))/n, one obtains a sequence converging to zero. REFERENCES J. M. De Koninck, N. Doyon and I. Katai, On the counting function for the Niven numbers, Acta Arith. 106 (2003), 265--275. E. J. Ionascu, H. Fredricksen, F. Luca and P. Stanica, Minimal Niven numbers, Acta Arith. 132 (2008), 135--159. E. J. Ionascu, F. Luca, P. Stanica and H. Fredricksen, Remarks on a sequence of minimal Niven numbers, Proceedings of the International Workshop, SSC 2007 (Sequences, Subsequences and Consequences) Springer, 162--168. I. Vardi, Niven numbers, Computational Recreations in Mathematics, Addison-Wesley, 1991, pp. 19 and 28--31. LINKS Eugen J. Ionascu and Ray Chandler, Table of n, a(n) for n = 1..2024 E. J. Ionascu, H. Fredricksen, F. Luca and P. Stanica, Minimal Niven numbers FORMULA a(n) =2^n-1 if n=2^k or a(n)=(2^(n+k-1)+2^n-2^(n-k)-1)/n if n=2^k-1 is a prime number; unknown for other values of n. EXAMPLE Example: If n=7 then 7(89)=623 which written in base 2 is 1001101111 using exactly 7 ones and 89 is the smallest positive integer with this property. Hence a(7)=89. The number 1001101111 is usually known as Niven number in base 2. We called 623 a minimal Niven number. MAPLE with(numtheory): fjv6:=proc(n, m) local i, j, k, l, x, x1, y, y1, z, z1, w, stopp, s, t, u, v, A, F, G, out; i:=n; stopp:=0; x1:=2^(m*i+6)-1; x:=x1 mod i; j:=0; while stopp=0 and j<=m*i+5 do l:=j; while stopp=0 and l<=m*i+4 do k:=l; while stopp=0 and k<=m*i+3 do s:=k; while stopp=0 and s<=m*i+2 do t:=s; while stopp=0 and t<=m*i+1 do v:=t; while stopp=0 and v<=m*i do y1:=2^(m*i+5-j)+2^(m*i+4-l)+2^(m*i+3-k)+2^(m*i+2-s)+2^(m*i+1-t)+2^(m*i-v); y:=y1 mod i; if y=x then z:=(x1-y1)/i; out:=[m*i, z]; stopp:=1; fi; v:=v+1; od; t:=t+1; od; s:=s+1; od; k:=k+1; od; l:=l+1; od; j:=j+1; od; if stopp=0 then out:=[m*i, 0]; fi; out; end: formula:=proc(n) local x, y, B, expon, outputis, theOddFactor; x:=n+1; B:=ifactors(x); expon:=B[2][1][2]; theOddFactor:=(n+1)/2^expon; y:=isprime(n); if theOddFactor=1 and y=true then outputis:=[n, (2^(n+expon-1)+2^n-2^(n-expon)-1)/n]; fi; if theOddFactor>1 or y=false then outputis:=fjv6(n, 1); fi; lprint(outputis[1], outputis[2]); end: fjfromis6:=proc(n, m) local k, B, expon, theoddfac, par, stopp, av, sub; av:=0; for k from n to m do par:=k mod 2; if par=0 then B:=ifactors(k); expon:=B[2][1][2]; theoddfac:=k/2^expon; sub:=fjv6(theoddfac, 2^expon); lprint(sub[1], sub[2]); fi; stopp:=0; if par=1 then formula(k); fi; od; end: fjfromis6(1, 185); PROG (PARI) a(n)=my(K=n); while(hammingweight(K)!=n, K+=n); K/n \\ Charles R Greathouse IV, Feb 04 2013 CROSSREFS Cf. A005349, A143115. Sequence in context: A154883 A109732 A114396 * A086517 A152677 A135374 Adjacent sequences:  A102029 A102030 A102031 * A102033 A102034 A102035 KEYWORD nonn AUTHOR Eugen J. Ionascu, Aug 03 2007 EXTENSIONS Edited by Ray Chandler, Nov 16 2008 STATUS approved

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