A353383 Irregular triangle T(n,k) with row n listing A003586(j) not divisible by 12 such that A352072(A003586(j)) = n.

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

Offset: 0

Author

Michael De Vlieger, Apr 15 2022

Keywords

• nonn
• easy
• base
• tabf

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.

Examples

			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.
		

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

• Michael De Vlieger, Table of n, a(n) for n = 0..6643
• Eric Weisstein's World of Mathematics, Duodecimal.
• Wikipedia, Duodecimal.

Crossrefs

Cf. A003586, A352072.

Programs

Mathematica

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

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.