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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 (list; graph; refs; listen; history; text; internal format)
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, 144-145.
LINKS
Eric Weisstein's World of Mathematics, Duodecimal.
Wikipedia, Duodecimal.
FORMULA
Row 0 contains the empty product, thus row length = 1.
Row n sorts {2^(2n-1), 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
Sequence in context: A325046 A160256 A151545 * A097274 A322572 A254438
KEYWORD
nonn,easy,base,tabf
AUTHOR
Michael De Vlieger, Apr 15 2022
STATUS
approved

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Last modified February 21 16:34 EST 2024. Contains 370237 sequences. (Running on oeis4.)