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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}.

%I A144912 #28 Feb 16 2025 08:33:09
%S A144912 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,
%T A144912 -12,0,-4,0,2,-2,-6,-10,-14,0,-2,2,-4,0,-4,-8,-12,-16,2,0,4,-2,2,-2,
%U A144912 -6,-10,-14,-18,0,-2,0,0,-6,0,-4,-8,-12,-16,-20
%N 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}.
%C A144912 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.
%C A144912 dm_num(b, n) = d(b - 1) iff all the digits in n are b - 1.
%C A144912 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).
%C A144912 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).
%C A144912 Defining m = ceiling((n + 1) / 2):
%C A144912 dm_num(b, n) = dm_num(b - 1, n) - 4 for b in [m + 1, n].
%C A144912 dm_num(m, n) = 0 for even n and 2 for odd n.
%C A144912 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, ...}.
%C A144912 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".
%H A144912 Reikku Kulon, <a href="/A144912/b144912.txt">Rows of triangle for b in [2, 142]</a>
%H A144912 Reikku Kulon, <a href="/A144912/a144912.c">C99 source to produce the triangle</a>
%H A144912 Reikku Kulon, <a href="/A144912/a144912.png">Triangle as 2048x2048 PNG image</a> (zero is white, primes are black and darker grays indicate fewer prime factors)
%H A144912 Reikku Kulon, <a href="/A144912/a144912_shadow.png">Triangle as 2048x2048 PNG image, extended to b in [2, 2 * n + 1]</a>
%H A144912 Reikku Kulon, <a href="/A144912/a144912_primes.png">Prime band as 16384x256 PNG image</a> (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)
%H A144912 Reikku Kulon, <a href="/A144912/a144912_primes_64k.png">Prime band as 16384x256 PNG image, starting from n = 57344</a>
%H A144912 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RedhefferMatrix.html">Redheffer Matrix</a>
%e A144912 Triangle begins:
%e A144912    0;
%e A144912    2, -2;
%e A144912   -1,  0, -4;
%e A144912    1,  2, -2, -6;
%e A144912    1,  0,  0, -4, -8;
%e A144912    3,  2,  2, -2, -6, -10;
%e A144912    ...
%t A144912 dmnum[b_,n_]:=2Total[IntegerDigits[n,b]]-(b-1)Floor[Log[b,n*b]]; (* after _Jinyuan Wang_ *)
%t A144912 Table[dmnum[b,n],{n,2,10},{b,2,n}] (* _Paolo Xausa_, Sep 26 2023 *)
%o A144912 (PARI) dm(b, n) = 2*sumdigits(n, b) - (b-1)*logint(n*b, b); \\ _Jinyuan Wang_, Jul 21 2020
%Y A144912 Cf. A002321, A031443, A083058, A144777, A144798, A144799, A144800, A144801, A144812, A144923, A240236.
%K A144912 sign,base,easy,frac,tabl
%O A144912 2,2
%A A144912 _Reikku Kulon_, Sep 25 2008, Oct 03 2008