A353385 Irregular triangle T(n,k) with row n listing A051037(j) not divisible by 60 such that A352219(j) = n.

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 8, 9, 16, 18, 24, 25, 36, 40, 45, 48, 50, 72, 75, 80, 90, 100, 144, 150, 200, 225, 400, 450, 27, 32, 54, 64, 96, 108, 125, 135, 160, 192, 216, 250, 270, 288, 320, 375, 432, 500, 576, 675, 750, 800, 864, 1000, 1125, 1350, 1600

Offset: 0

Author

Michael De Vlieger, Apr 15 2022

Keywords

• nonn
• easy
• base
• tabf

Comments

All terms in A051037 are products T(n,k)*60^j, j >= 0.

When expressed in base 60, T(n,k) does not end in zero, yet 1/T(n,k) is a terminating fraction, regular to 60.

The first 11 terms are the proper divisors of 60.

For these reasons, the terms may be called sexagesimal "proper regular" numbers.

Examples

			For row w, plot terms m = 2^x * 3^y * 5^z at (x,y,z). Rows are labeled below the figures parenthetically for clarity. The x axis points toward the bottom right, the y axis to the bottom left, and the z axis upward. In the plot, we mark terms from previous rows by ".", and use "*" to show the origin, that is, the empty product 1:
                                                   125
                                                375   250
                                             1125   750   500
                                         3375  2250        1000
                                            6750              2000
                         25                          .           4000
                      75    50                    .     .           8000
                  225   150   100              .     .     .
                     450         200      675     .           .
                                    400     1350                 .
         5                .                          .            800
     15    10          .     .                    .     .           1600
         30    20   45     .     .              .     .     .
                      90          40      135     .           .
                                      80      270                 .
1         *                *                          *            160
       3     2          .     .                    .     .            320
          6     4    9     .     .              .     .     .
            12         18     .     8       27     .     .     .
                          36    24    16       54     .     .     .
                             72    48            108     .     .    32
                               144                  216     .    96    64
                                                       432   288   192
                                                          864   432
                                                            1728
(0)      (1)              (2)                        (3)
The terms in row w are sorted, hence row 1 has {2, 3, 4, 5, 6, 10, 12, 15, 20, 30}.
		

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..10250 (rows n = 1..40, flattened)
• Eric Weisstein's World of Mathematics, Sexagesimal
• Wikipedia, Regular number.

Crossrefs

Cf. A017641, A051037, A352219.

Programs

Mathematica

Block[{t, s = DeleteCases[Sort[Flatten[Table[{2^a* 3^b * 5^c, Max[Ceiling[a/2], b, c]}, {a, 0, Log2[#]}, {b, 0, Log[3, #/(2^a)]}, {c, 0, Log[5, #/(2^a*3^b)]}], 2]] &[60^3], _?(Mod[First[#], 60] == 0 &)]}, #[[1 ;; 2 + LengthWhile[Rest@ Differences[Length /@ #], # == 12 &]]] &@ Map[s[[#, 1]] &, Values@ PositionIndex[s[[All, -1]]]]] // Flatten

Formula

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

For n > 0, length of row n = 12(n-1) + 10 = A017641(n-1).