

A353383


Irregular triangle T(n,k) with row n listing A003586(j) not divisible by 12 such that A352072(A003586(j)) = n.


1



1, 2, 3, 4, 6, 8, 9, 16, 18, 27, 32, 54, 64, 81, 128, 162, 256, 243, 486, 512, 1024, 729, 1458, 2048, 4096, 2187, 4374, 8192, 16384, 6561, 13122, 32768, 65536, 19683, 39366, 131072, 262144, 59049, 118098, 524288, 1048576, 177147, 354294, 2097152, 4194304, 531441, 1062882, 8388608, 16777216
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OFFSET

0,2


COMMENTS

All terms in A003586 are products T(n,k)*12^j, j >= 0.
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.
The first 5 terms are the proper divisors of 12.
For these reasons, the terms may be called duodecimal "proper regular" numbers.


REFERENCES

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, 144145.


LINKS

Eric Weisstein's World of Mathematics, Duodecimal.


FORMULA

Row 0 contains the empty product, thus row length = 1.
Row n sorts {2^(2n1), 3^n, 2^(2n), 2*3^n}, thus row length = 4.


EXAMPLE

Row 0 contains 1 since 1 is the empty product.
Row 1 contains 2, 3, 4, and 6 since these divide 12.
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.
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.


MATHEMATICA

{{1}}~Join~Array[Union@ Flatten@ {#, 2 #} &@ {2^(2 #  1), 3^#} &, 12] // Flatten


CROSSREFS



KEYWORD

nonn,easy,base,tabf


AUTHOR



STATUS

approved



