User:Enrique Pérez Herrero/Salad

FORMULAE SALAD

This is compilations of sequences that I egoistically need for other purposes, and it is very handy to have them already written somewhere:

NUMBER OF DIGITS

b	Sequence	Description
2	A029837	Binary order of n: log_2(n) rounded up to next integer.
2	A070939	Length of binary representation of n.
3	A081604	String-length of ternary representation of n.
4	A110591	String length of base 4 representation of n.
10	A055642	Number of digits in decimal expansion of n.

The expression can be extended to any real number $x$:

$\displaystyle D_{b}(x)=1;|x|<1;$

$\displaystyle D_{b}(x)=1+\lfloor \log _{b}|x|\rfloor ;$ $|x|\geq 1$

[1]

[2]

Mathematica Programming

(* NUMBER OF DIGITS related OEIS sequences *)
Digits[x_,b_Integer:10]/;If[TrueQ[b > 1], True, Message[Digits::ibase, b]; False]:=Max[1+Floor[Log[b,Abs[x]]],1];
Digits::ibase="Base `1` is not an integer greater than 1";
A029837[n_]:=Ceiling[Log[2,n]];
A070939[n_]:=Digits[n,2];
A081604[n_]:=Digits[n,3];
A110591[n_]:=Digits[n,4];
A055642[n_]:=Digits[n];

(*Adding help*)
Digits::usage="Digits[x] gives the number of digits in the base 10 representation of the integer part of the real number x.
Digits[x,b] gives the number of digits in the base b representation of the integer part of x.";
A029837::usage="A029837: Binary order of n: log_2(n) rounded up to next integer.";
A055642::usage="A055642: Number of digits in decimal expansion of n.";
A070939::usage="A070939: Length of binary representation of n.";
A081604::usage="A081604: String-length of ternary representation of n.";
A110591::usage="A110591: String length of base 4 representation of n. String length of A007090.";

k*n digit-reversed mod k

k	Sequence	Description
2	A084052	2*n digit-reversed mod 2
3	A000004	3*n digit-reversed mod 3
4	A084053	4*n digit-reversed mod 4
5	A084054	5*n digit-reversed mod 5
6	A084055	6*n digit-reversed mod 6
7	A084339	7*n digit-reversed mod 7
8	A084340	8*n digit-reversed mod 8
9	A000004	9*n digit-reversed mod 9
10	A000030	10*n digit-reversed mod 10
13	A084341	13*n digit-reversed mod 13

Formulae for k*n digit-reversed mod k

Initial digit of n: A000030

Mathematica Programming:

(*Adding help*)
A000030::usage="A000030: Initial digit of n.";
A055642::usage="A055642: Number of digits in decimal expansion of n.";