A105116 -id:A105116 - cp's OEIS frontend

A105424 The part of n in base phi left of the decimal point, using a greedy algorithm representation (more precisely, using the Bergman-canonical representation).

0, 1, 10, 100, 101, 1000, 1010, 10000, 10001, 10010, 10100, 10101, 100000, 100010, 100100, 100101, 101000, 101010, 1000000, 1000001, 1000010, 1000100, 1000101, 1001000, 1001010, 1010000, 1010001, 1010010, 1010100, 1010101, 10000000

Offset: 0

Bryan Jacobs (bryanjj(AT)gmail.com), Apr 08 2005

			2 = 10.01 in base phi, so left of the decimal point is 10.
The first few numbers written in base phi:
0 = 0.
1 = 1.
2 = 10.01
3 = 100.01
4 = 101.01
5 = 1000.1001
6 = 1010.0001
7 = 10000.0001
8 = 10001.0001
9 = 10010.0101
10 = 10100.0101
11 = 10101.0101
12 = 100000.101001
13 = 100010.001001
14 = 100100.001001
15 = 100101.001001
16 = 101000.100001
17 = 101010.000001
18 = 1000000.000001
19 = 1000001.000001
20 = 1000010.010001
21 = 1000100.010001
22 = 1000101.010001
23 = 1001000.100101
24 = 1001010.000101
...

T. D. Noe, Table of n, a(n) for n = 0..1000
F. Michel Dekking, How to add two natural numbers in base phi, arXiv:2002.01665 [math.NT], 5 Feb 2020.
Ron Knott, Phigits and the Base Phi representation.
Ron Knott, Phigits and the Base Phi representation [Local copy, pdf only]
Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023. [Note that this document has been revised multiple times.]

Cf. A001622, A055778, A105425, A104605.
See A341722 for the part to the right of the decimal point.
Cf. A105116 (base e), A344939 (base Pi).

nn = 1000; len = 2*Ceiling[Log[GoldenRatio, nn]]; Table[d = RealDigits[n, GoldenRatio, len]; FromDigits[Take[d[[1]], d[[2]]]], {n, 0, nn}] (* T. D. Noe, May 20 2011 *)

Definition clarified by N. J. A. Sloane, May 27 2023

A344939 The part of n left of the radix point when written in base Pi using a greedy algorithm representation.

Original entry on oeis.org

0, 1, 2, 3, 10, 11, 12, 20, 21, 22, 100, 101, 102, 103, 110, 111, 112, 120, 121, 122, 200, 201, 202, 210, 211, 212, 213, 220, 221, 222, 300, 301, 1000, 1001, 1002, 1010, 1011, 1012, 1020, 1021, 1022, 1100, 1101, 1102, 1103, 1110, 1111, 1112, 1120, 1121, 1122

Offset: 0

Paolo Xausa, Jun 03 2021

			a(5) = 11 because 5 in base Pi is 11.22012202...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Non-integer base of numeration.

Cf. A000796 (Pi digits). Subsequence of A007090.
Cf. A105116 (base e), A105424 (base phi).
Cf. A366721 (number of digits).

A344939 := proc(n)
    local e,ntrunc,a,d;
    Digits := 1000 ;
    if n = 0 then
        return 0 ;
    end if;
    ntrunc := n ;
    e := floor(log(n)/log(Pi)) ;
    a := 0 ;
    while e >= 0 do
        d := floor(ntrunc/Pi^e) ;
        a := 10*a+d ;
        ntrunc := evalf(ntrunc-d*Pi^e) ;
        e := e-1 ;
    end do:
    a ;
end proc:
seq(A344939(n),n=0..15) ; # R. J. Mathar, Aug 16 2021

A344939[n_]:=FromDigits[First[RealDigits[n,Pi,Floor[Log[Pi,Max[n,1]]]+1]]];
Array[A344939,100,0] (* Paolo Xausa, Oct 17 2023 *)

Name edited by Paolo Xausa, Oct 18 2023

A363832 Number of digits left of the radix point of n when written in base e using a greedy algorithm representation.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 0

Paolo Xausa, Oct 19 2023

Essentially the same as A004233. - R. J. Mathar, Oct 23 2023

			a(10) = 3 because 10 in base e (102.11201...) has 3 digits before the radix point.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Non-integer base of numeration.

Cf. A001113, A055642, A105116, A362692, A366721.

A363832[n_]:=Floor[Log[E,Max[n,1]]]+1;Array[A363832,100,0]

a(0) = 1; for n >= 1, a(n) = floor(log_e(n)) + 1.

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A105424 The part of n in base phi left of the decimal point, using a greedy algorithm representation (more precisely, using the Bergman-canonical representation).

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A344939 The part of n left of the radix point when written in base Pi using a greedy algorithm representation.

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A363832 Number of digits left of the radix point of n when written in base e using a greedy algorithm representation.

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