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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}. 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
2,2
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".
LINKS
Reikku Kulon, Triangle as 2048x2048 PNG image (zero is white, primes are black and darker grays indicate fewer prime factors)
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)
Eric Weisstein's World of Mathematics, Redheffer Matrix
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]]-(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 *)
PROG
(PARI) dm(b, n) = 2*sumdigits(n, b) - (b-1)*logint(n*b, b); \\ Jinyuan Wang, Jul 21 2020
CROSSREFS
Sequence in context: A064044 A213980 A349373 * A306708 A145337 A171941
KEYWORD
sign,base,easy,frac,tabl
AUTHOR
Reikku Kulon, Sep 25 2008, Oct 03 2008
STATUS
approved

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Last modified February 28 22:27 EST 2024. Contains 370400 sequences. (Running on oeis4.)