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}.

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

Offset: 2

Author

Reikku Kulon, Sep 25 2008, Oct 03 2008

Keywords

• sign
• base
• easy
• frac
• tabl

Comments

The unreduced numerator of dm(b, n) is Sum_{i=1..d} (2*d_i - (b-1)), 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".

Examples

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

Links

• Reikku Kulon, Rows of triangle for b in [2, 142]
• Reikku Kulon, C99 source to produce the triangle
• Reikku Kulon, Triangle as 2048x2048 PNG image (zero is white, primes are black and darker grays indicate fewer prime factors)
• Reikku Kulon, Triangle as 2048x2048 PNG image, extended to b in [2, 2 * n + 1]
• Reikku Kulon, Prime band as 16384x256 PNG image (note the curves coincident with new strips of primes, as well as the second band which appears at 4096 and corresponds to the 637/638 gap in A031443)
• Reikku Kulon, Prime band as 16384x256 PNG image, starting from n = 57344
• Eric Weisstein's World of Mathematics, Redheffer Matrix

Crossrefs

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

Programs

Mathematica

dmnum[b_,n_]:=2Total[IntegerDigits[n,b]]-(b-1)Floor[Log[b,n*b]]; (* after Jinyuan Wang *)
Table[dmnum[b,n],{n,2,10},{b,2,n}] (* Paolo Xausa, Sep 26 2023 *)

PARI

dm(b, n) = 2*sumdigits(n, b) - (b-1)*logint(n*b, b); \\ Jinyuan Wang, Jul 21 2020