OFFSET
0,12
COMMENTS
The sign function is defined by:
- sign(0) = 0,
- sign(n) = +1 for any n > 0,
- sign(n) = -1 for any n < 0.
The graph of the sequence has some similarities with a Takagi (or blancmange) curve.
Visually, the sequence is of fractal nature; for k > 2, the scatterplot of the first 10^k terms is similar to that of the first 10^(k+1) terms.
We also have symmetries:
- for k = 1..6: let m_k = (10^k)/2-1: for i = 0..m_k, we have a(m_k - i) = a(m_k + i),
- this relation is conjectured to hold for any k > 0,
- this would be equivalent to saying that, for any k > 0 and i = 0..m_k, sign(A007953(5*(m_k - i)) - A007953(m_k - i)) = - sign(A007953(5*(m_k + i + 1)) - A007953(m_k + i + 1)).
For b > 1, let d_b be the digital sum in base b; in particular:
- d_2 = A000120,
- d_3 = A053735,
- d_10 = A007953.
For any b > 1 and n >= 0, d_b(b*n) = d_b(n).
Also, a(n) = Sum_{k=0..n} sign(d_10(5*k) - d_10(k)).
For b > 1, i > 0 and j > 0 such that neither i nor j are divisible by b, let F(b,i,j) be the function defined by n -> Sum_{k=0..n} sign(d_b(i*k) - d_b(j*k)); in particular:
- F(10,5,1) = a (this sequence).
Also, F(b,i,i) = 0 and F(b,i,j) = -F(b,j,i).
Conjecturally, we have three kinds of behaviors:
- if i = j, then F(b,i,j) = 0,
- otherwise if i and j divide b, then F(b,i,j) has infinitely many zeros (and infinitely many nonzero values), and has similar fractal nature and exhibits similar symmetries as the present sequence,
- otherwise |F(b,i,j)| tends to infinity (and has only a finite number of zeros).
a(n) = 0 for n = 0, 2, 4, 6, 8, 9, 89, 90, 92, 94, 96, 98, 99, 899, 900, 902, 904, 906, 908, 909, 989, 990, 992, 994, 996, 998, 999, 8999, ...
LINKS
Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Scatterplot of F(10,5,1) (this sequence)
Rémy Sigrist, Scatterplot of F(10,2,1)
Rémy Sigrist, Scatterplot of F(10,5,2)
Rémy Sigrist, Scatterplot of F(10,7,1)
Rémy Sigrist, Scatterplot of F(18,6,3)
Rémy Sigrist, Scatterplot of F(42,7,2)
EXAMPLE
The first terms, alongside the digital sum of 5*n and n, and the sign of their difference, are:
n a(n) d_10(5*n) d_10(n) sign
-- ---- --------- ------- ----
0 0 0 0 0
1 1 5 1 +1
2 0 1 2 -1
3 1 6 3 +1
4 0 2 4 -1
5 1 7 5 +1
6 0 3 6 -1
7 1 8 7 +1
8 0 4 8 -1
9 0 9 9 0
10 1 5 1 +1
11 2 10 2 +1
12 3 6 3 +1
13 4 11 4 +1
14 5 7 5 +1
15 6 12 6 +1
16 7 8 7 +1
17 8 13 8 +1
18 8 9 9 0
19 9 14 10 +1
20 8 1 2 -1
21 9 6 3 +1
22 8 2 4 -1
23 9 7 5 +1
24 8 3 6 -1
25 9 8 7 +1
MATHEMATICA
With[{s = Table[Total@ IntegerDigits[5 k] - Total@ IntegerDigits@ k, {k, 0, 76}]}, Table[Total@ Map[Sign, Take[s, n]], {n, Length@ s}]] (* Michael De Vlieger, Jul 20 2017 *)
PROG
(PARI) a(n) = sum(k=0, n, sign(sum digits(5*k) - sum digits(k)))
(Python)
from sympy import sign
from sympy.ntheory.factor_ import digits
def a(n): return sum([sign(sum(digits(5*k)[1:]) - sum(digits(k)[1:])) for k in range(n + 1)])
print([a(n) for n in range(51)]) # Indranil Ghosh, Aug 02 2017
CROSSREFS
KEYWORD
AUTHOR
Rémy Sigrist, Jul 18 2017
STATUS
approved