

A144912


Unreduced numerators of digital mean, dm_num(b, n), with rows n in {2, 3, 4, ...} and columns b in {2, 3, 4, ..., n}.


11



0, 2, 2, 1, 0, 4, 1, 2, 2, 6, 1, 0, 0, 4, 8, 3, 2, 2, 2, 6, 10, 2, 4, 2, 0, 4, 8, 12, 0, 4, 0, 2, 2, 6, 10, 14, 0, 2, 2, 4, 0, 4, 8, 12, 16, 2, 0, 4, 2, 2, 2, 6, 10, 14, 18, 0, 2, 0, 0, 6, 0, 4, 8, 12, 16, 20
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OFFSET

2,2


COMMENTS

The unreduced numerator of dm(b, n) is Sum_{i=1..d} (2*d_i  (b1)), where d is the number of digits in the base b representation of n and d_i the individual digits. The corresponding denominator is 2 * d, giving a value in ((b  1) / 2, (b  1) / 2] for n > 0.
dm_num(b, n) = d(b  1) iff all the digits in n are b  1.
dm_num(b, n) = 2(b  2) for b = n, because n in base n is 10, giving dm_num(n, n) = 2  n + 1 + 0  n + 1 = 4  2 * n = 2(n  2).
dm_num(b, n) = 0 for odd b and n having all digits equal to (b  1) / 2, as well as for many other (b, n).
Defining m = ceiling((n + 1) / 2):
dm_num(b, n) = dm_num(b  1, n)  4 for b in [m + 1, n].
dm_num(m, n) = 0 for even n and 2 for odd n.
dm_num(m  1, n) = 6  n for even n > 4 and 9  n for odd n > 5, producing a sequence of first differences {+2, 4, +2, 4, ...}.
Triangular patterns become clearly visible for large n, defined by additive periodicities along rational slopes. Zeros along the triangle borders correspond to ones in the Redheffer matrix until odd values become dominant. The line along m is the border between the two largest triangles. This pattern is masked by aliasing effects for small bases, notably including base 10, due to the thinness of the triangles which dominate at small b. Odd values may represent "artifacts" caused by "interference".


LINKS



EXAMPLE

Triangle begins:
0;
2, 2;
1, 0, 4;
1, 2, 2, 6;
1, 0, 0, 4, 8;
3, 2, 2, 2, 6, 10;
...


MATHEMATICA

dmnum[b_, n_]:=2Total[IntegerDigits[n, b]](b1)Floor[Log[b, n*b]]; (* after Jinyuan Wang *)
Table[dmnum[b, n], {n, 2, 10}, {b, 2, n}] (* Paolo Xausa, Sep 26 2023 *)


PROG

(PARI) dm(b, n) = 2*sumdigits(n, b)  (b1)*logint(n*b, b); \\ Jinyuan Wang, Jul 21 2020


CROSSREFS

Cf. A002321, A031443, A083058, A144777, A144798, A144799, A144800, A144801, A144812, A144923, A240236.


KEYWORD



AUTHOR



STATUS

approved



