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%I #15 Mar 15 2020 05:10:30
%S 12,13,14,15,28,29,30,31,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,
%T 59,60,61,62,63,76,77,78,79,92,93,94,95,108,109,110,111,112,113,114,
%U 115,116,117,118,119,120,121,122,123,124,125,126,127,140,141,142,143
%N Positive integers n where the number of parts function on the set of 4-ary partitions of n is equidistributed mod 4.
%C An integer n is in the list if and only if n_i=3 for some index i>0 where n = Sum_{i>=0}n_i4^i is the base 4 representation of n.
%H Tom Edgar, <a href="http://arxiv.org/abs/1603.00085">The distribution of the number of parts of m-ary partitions modulo m.</a>, arXiv:1603.00085 [math.CO], 2016.
%e There are four 4-ary partitions of 12: one has 12 parts (1+1+1+1+1+1+1+1+1+1+1+1), one has 3 parts (4+4+4), one has 9 parts (4+1+1+1+1+1+1+1+1), and one has 6 parts (4+4+1+1+1+1); thus, modulo 4, the number of parts function is equidistributed mod 4 and so 12 is a term.
%e There are six 4-ary partitions of 16 so the number of parts function cannot be equidistributed mod 4. Thus, 16 is not a term.
%o (Sage) [n for n in [1..150] if 3 in n.digits(4)[1:]]
%Y Cf. A005705, A272344.
%K nonn
%O 1,1
%A _Tom Edgar_, Apr 28 2016
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