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%I #10 Apr 21 2022 09:15:18
%S 1,2,3,4,6,8,9,16,18,27,32,54,64,81,128,162,256,243,486,512,1024,729,
%T 1458,2048,4096,2187,4374,8192,16384,6561,13122,32768,65536,19683,
%U 39366,131072,262144,59049,118098,524288,1048576,177147,354294,2097152,4194304,531441,1062882,8388608,16777216
%N Irregular triangle T(n,k) with row n listing A003586(j) not divisible by 12 such that A352072(A003586(j)) = n.
%C All terms in A003586 are products T(n,k)*12^j, j >= 0.
%C When expressed in base 12, T(n,k) does not end in zero, yet 1/T(n,k) is a terminating fraction, regular to 12.
%C The first 5 terms are the proper divisors of 12.
%C For these reasons, the terms may be called duodecimal "proper regular" numbers.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Chapter IX: The Representation of Numbers by Decimals, Theorem 136. 8th ed., Oxford Univ. Press, 2008, 144-145.
%H Michael De Vlieger, <a href="/A353383/b353383.txt">Table of n, a(n) for n = 0..6643</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Duodecimal.html">Duodecimal</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Duodecimal">Duodecimal</a>.
%F Row 0 contains the empty product, thus row length = 1.
%F Row n sorts {2^(2n-1), 3^n, 2^(2n), 2*3^n}, thus row length = 4.
%e Row 0 contains 1 since 1 is the empty product.
%e Row 1 contains 2, 3, 4, and 6 since these divide 12.
%e Row 2 contains 8, 9, 16, and 18 since these divide 12^2 but not 12. The other divisors of 12^2 either divide smaller powers of 12 or they are divisible by 12 and do not appear.
%e Row 3 contains 27, 32, 54, and 64 since these divide 12^3 but not 12^2. The other divisors of 12^3 either divide smaller powers of 12 or they are divisible by 12 therefore do not appear.
%t {{1}}~Join~Array[Union@ Flatten@ {#, 2 #} &@ {2^(2 # - 1), 3^#} &, 12] // Flatten
%Y Cf. A003586, A352072.
%K nonn,easy,base,tabf
%O 0,2
%A _Michael De Vlieger_, Apr 15 2022
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