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

0,2

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

Michael De Vlieger, Table of n, a(n) for n = 0..6643

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

Cf. A003586, A352072.

Sequence in context: A325046 A160256 A151545 * A097274 A322572 A254438

Adjacent sequences: A353380 A353381 A353382 * A353384 A353385 A353386

Keyword

nonn,easy,base,tabf

Author

Michael De Vlieger, Apr 15 2022

Status

approved