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A353384 Irregular triangle T(n,k) with row n listing A003592(j) not divisible by 20 such that A352218(A003592(j)) = n.

%I A353384 #9 Apr 21 2022 09:15:22
%S A353384 1,2,4,5,10,8,16,25,50,32,64,125,250,128,256,625,1250,512,1024,3125,
%T A353384 6250,2048,4096,15625,31250,8192,16384,78125,156250,32768,65536,
%U A353384 390625,781250,131072,262144,1953125,3906250,524288,1048576,9765625,19531250,2097152,4194304,48828125,97656250
%N A353384 Irregular triangle T(n,k) with row n listing A003592(j) not divisible by 20 such that A352218(A003592(j)) = n.
%C A353384 All terms in A003592 are products T(n,k)*20^j, j >= 0.
%C A353384 When expressed in base 20, T(n,k) does not end in zero, yet 1/T(n,k) is a terminating fraction, regular to 20.
%C A353384 The first 5 terms are the proper divisors of 20.
%C A353384 For these reasons, the terms may be called vigesimal "proper regular" numbers.
%D A353384 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 A353384 Michael De Vlieger, <a href="/A353384/b353384.txt">Table of n, a(n) for n = 0..5722</a>
%H A353384 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Vigesimal.html">Vigesimal</a>
%H A353384 Wikipedia, <a href="https://en.wikipedia.org/wiki/Vigesimal">Vigesimal</a>
%F A353384 Row 0 contains the empty product, thus row length = 1.
%F A353384 Row n sorts {2^(2n-1), 5^n, 2^(2n), 2*5^n}, thus row length = 4.
%e A353384 Row 0 contains 1 since 1 is the empty product.
%e A353384 Row 1 contains 2, 4, 5, and 10 since these divide 20 and are not divisible by 20.
%e A353384 Row 2 contains 8, 16, 25, and 50 since these divide 20^2 but not 20. The other divisors of 20^2 either divide smaller powers of 20 or they are divisible by 20 and do not appear.
%e A353384 Row 3 contains 32, 64, 125, and 250 since these divide 20^3 but not 20^2. The other divisors of 20^3 either divide smaller powers of 20 or they are divisible by 20 therefore do not appear.
%t A353384 {{1}}~Join~Array[Union@ Flatten@ {#, 2 #} &@ {2^(2 # - 1), 5^#} &, 11] // Flatten
%Y A353384 Cf. A003592, A352218.
%K A353384 nonn,easy,base,tabf
%O A353384 0,2
%A A353384 _Michael De Vlieger_, Apr 15 2022