A362692 -id:A362692 - cp's OEIS frontend

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

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.

A366721 Number of digits left of the radix point of n when written in base Pi using a greedy algorithm representation.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Oct 17 2023

More than the usual number of terms are shown to distinguish this sequence from A185679.

			a(10) = 3 because 10 in base Pi (100.01022...) 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. A000796, A055642, A185679, A344939, A362692, A363832.

A366721[n_]:=Floor[Log[Pi,Max[n,1]]]+1;Array[A366721,200,0]

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

A362872 Length of the "fractional part" of the phi-representation of n.

Original entry on oeis.org

0, 0, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10, 10

Offset: 0

Jeffrey Shallit, May 07 2023

nonn

The phi-representation of n is the (essentially) unique way to write n = Sum_{j=L..R} b(j)*phi^j, where b(j) is in {0,1} and -oo < L <= 0 <= R, where phi = (1+sqrt(5))/2, subject to the condition that b(j)b(j+1) != 1. The "fractional" part is the string of bits b(L)b(L+1)...b(-1), and its length is thus L.

The gaps between consecutive terms are all either 0 or 2, and a gap of 2 occurs if and only if n = L(2i+1) for i >= 0. This is equivalent to Theorem 2.1 of Sanchis and Sanchis (2001).

			The phi-representation of 20 is 1000010.010001, so a(20) = 6.

George Bergman, A number system with an irrational base, Math. Mag. 31 (1957), 98-110.
G. R. Sanchis and L. A. Sanchis, On the frequency of occurrence of α^i in the α-expansions of the positive integers, Fibonacci Quart. 39 (2001), 123-137.

Cf. A341722, A362692.

There is a linear representation of rank 11 for a(n).

A371672 a(n) = floor(log_phi(n)) with phi = A001622.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 1

Aleksandar V. Petojevic, Apr 02 2024

nonn

Aleksandar Petojević, Marjana Gorjanac Ranitović, Dragan Rastovac, and Milinko Mandić, The Golden Ratio, Factorials, and the Lambert W Function, Journal of Integer Sequences, Vol. 27 (2024), Article 24.5.7. See p. 4.

Cf. A000195, A000523, A001622, A004216, A362692.

Table[IntegerPart[Log[n]/Log[GoldenRatio]],{n,1,130}]

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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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A366721 Number of digits left of the radix point of n when written in base Pi using a greedy algorithm representation.

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A362872 Length of the "fractional part" of the phi-representation of n.

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A371672 a(n) = floor(log_phi(n)) with phi = A001622.

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