A276134 a(5n) = a(n), a(5n+1) = a(5n+2) = a(5n+3) = a(5n+4) = a(n) + 1, a(0) = 0.

0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 2

Offset: 0

Author

Ilya Gutkovskiy, Aug 21 2016

Keywords

• nonn
• base
• look

Comments

Number of nonzero digits in the base 5 representation of n.

Fixed point of the mapping 0 -> 01111, 1 -> 12222, 2 -> 23333, ...

Self-similar or fractal sequence (underlining every fifth term, reproduce the original sequence).

Examples

			The evolution starting with 0 is: 0 -> 01111 -> 0111112222122221222212222 -> ...
...
a(0) = 0;
a(1) = a(5*0+1) = a(0) + 1 = 1;
a(2) = a(5*0+2) = a(0) + 1 = 1;
a(3) = a(5*0+3) = a(0) + 1 = 1;
a(4) = a(5*0+4) = a(0) + 1 = 1;
a(5) = a(5*1+0) = a(1) = 1;
a(6) = a(5*1+1) = a(1) + 1 = 2, etc.
...
Also a(10) = 1, because 10 (base 10) = 20 (base 5) and 20 has 1 nonzero digit.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Michael Gilleland, Some Self-Similar Integer Sequences
• Ilya Gutkovskiy, Illustration (mapping 0 -> 01111, 1 -> 12222, 2 -> 23333, ...)

Crossrefs

Cf. A000034, A000120, A093178, A160384, A160385.

Programs

Maple

f:= n -> nops(subs(0=NULL,convert(n,base,5))):
map(f, [$0..100]); # Robert Israel, Sep 07 2016

Mathematica

Join[{0}, Table[IntegerLength[n, 5] - DigitCount[n, 5, 0], {n, 120}]]

Formula

a(5^k) = 1.

a(5^k-1) = k.

a(5^k-m) = k, k>0, m = 2,3,4.

a(5^k+m) = 2, k>0, m = 1,2,3,4.

a(5^k-a(5^k)) = k.

a(5^k+(-1)^k) = (k + (-1)^k*(k - 1) + 3)/2.

a(5^k+(-1)^k-1) = A093178(k).

a(5^k+(-1)^k+1) = A000034(k+1), k>0.

G.f. g(x) satisfies g(x) = (1+x+x^2+x^3+x^4)*g(x^5) + (x+x^2+x^3+x^4)/(1-x^5). - Robert Israel, Sep 07 2016