OFFSET
0,2
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.
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.
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).
EXAMPLE
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}.
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
CROSSREFS
KEYWORD
nonn,easy,base,tabf
AUTHOR
Michael De Vlieger, Apr 15 2022
STATUS
approved