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A256785 Numbers n such that digitsum(n) is a whole number when n is represented in the fractional base 1.5 = 3/2.

%I A256785 #15 Mar 12 2019 18:33:58
%S A256785 1,5,11,14,20,21,22,23,26,29,30,31,33,34,38,39,40,41,45,46,51,52,53,
%T A256785 56,57,58,60,61,65,69,70,71,74,78,79,83,84,85,87,88,89,90,91,95,101,
%U A256785 105,106,110,111,112,113,116,117,118,122,126,127,132,133,135,136,140,146,149,155,159,160,161,164,165,166,168,169,173,174,175
%N A256785 Numbers n such that digitsum(n) is a whole number when n is represented in the fractional base 1.5 = 3/2.
%C A256785 Base 1.5 requires three digits: 1, 0 and H = 0.5. For example:
%C A256785 1 = 1 = 1 * 1.5^0
%C A256785 2 = 1H = 1 * 1.5^1 + 0.5 * 1.5^0 = 1.5 + 0.5
%C A256785 3 = 1H0 = 1 * 1.5^2 + 0.5 * 1.5^1 = 2.25 + 0.75
%C A256785 4 = 1H1 = 1 * 1.5^2 + 0.5 * 1.5^1 + 1 * 1.5^0 = 2.25 + 0.75 + 1
%C A256785 5 = 1H0H = 1 * 1.5^3 + 0.5 * 1.5^2 + 0.5 * 1.5^0 = 3.375 + 1.125 + 0.5
%C A256785 6 = 1H10 = 1 * 1.5^3 + 0.5 * 1.5^2 + 1 * 1.5^1 = 3.375 + 1.125 + 1.5
%C A256785 7 = 1H11 = 1 * 1.5^3 + 0.5 * 1.5^2 + 1 * 1.5^1 + 1 * 1.5^0 = 3.375 + 1.125 + 1.5 + 1
%C A256785 The sequence above lists the n for which digsum(n,base=1.5) is a whole number.
%H A256785 Anthony Sand, <a href="/A256785/b256785.txt">Table of n, a(n) for n = 1..1000</a>
%H A256785 Matvey Borodin, Hannah Han, Kaylee Ji, Tanya Khovanova, Alexander Peng, David Sun, Isabel Tu, Jason Yang, William Yang, Kevin Zhang, Kevin Zhao, <a href="https://arxiv.org/abs/1901.09818">Variants of Base 3 over 2</a>, arXiv:1901.09818 [math.NT], 2019
%e A256785 The sequence begins with 1, 5 and 11, because:
%e A256785 digsum(1,b=1.5) = 1
%e A256785 digsum(5,b=1.5) = 2 = digsum(1H0H) = 1 + 0.5 + 0.5
%e A256785 digsum(11,b=1.5) = 4 = digsum(1H11H) = 1 + 0.5 + 1 + 1 + 0.5
%e A256785 The digsums are all whole numbers. However, 2, 3 and 4 are excluded because:
%e A256785 digsum(2,b=1.5) = 1.5 = digsum(1H) = 1 + 0.5
%e A256785 digsum(3,b=1.5) = 1.5 = digsum(1H0) = 1 + 0.5 + 0
%e A256785 digsum(4,b=1.5) = 2.5 = digsum(1H1) = 1 + 0.5 + 1
%e A256785 The digsums are not whole numbers.
%o A256785 (PARI) { b=3/2; dmx=30; d=vector(dmx); nmx=1000; n=0; ni=0; while(ni<nmx, n++; nn=n; di=0; while(nn>0, di++; d[di]=nn-floor(nn/b)*b; nn\=b; ); s=0; for(i=1,di,s+=d[i]); if(floor(s)==s, ni++; write("digsum.txt",ni," ",n)); ); }
%Y A256785 Cf. A007953, A054683, A054684.
%K A256785 nonn,base
%O A256785 1,2
%A A256785 _Anthony Sand_, Apr 10 2015