A353384 - OEIS

%I #9 Apr 21 2022 09:15:22

%S 1,2,4,5,10,8,16,25,50,32,64,125,250,128,256,625,1250,512,1024,3125,

%T 6250,2048,4096,15625,31250,8192,16384,78125,156250,32768,65536,

%U 390625,781250,131072,262144,1953125,3906250,524288,1048576,9765625,19531250,2097152,4194304,48828125,97656250

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

%C All terms in A003592 are products T(n,k)*20^j, j >= 0.

%C 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 The first 5 terms are the proper divisors of 20.

%C For these reasons, the terms may be called vigesimal "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="/A353384/b353384.txt">Table of n, a(n) for n = 0..5722</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Vigesimal.html">Vigesimal</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Vigesimal">Vigesimal</a>

%F Row 0 contains the empty product, thus row length = 1.

%F Row n sorts {2^(2n-1), 5^n, 2^(2n), 2*5^n}, thus row length = 4.

%e Row 0 contains 1 since 1 is the empty product.

%e Row 1 contains 2, 4, 5, and 10 since these divide 20 and are not divisible by 20.

%e 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 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 {{1}}~Join~Array[Union@ Flatten@ {#, 2 #} &@ {2^(2 # - 1), 5^#} &, 11] // Flatten

%Y Cf. A003592, A352218.

%K nonn,easy,base,tabf

%O 0,2

%A _Michael De Vlieger_, Apr 15 2022