A032634 -id:A032634 - cp's OEIS frontend

A081573 Decimal expansion of Sum_(1/(2^q-1)) with the summation extending over all pairs of integers gcd(p,q) = 1, 0 < p/q < e = 2.718... .

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

Offset: 1

Benoit Cloitre, Apr 21 2003

			4.866141732...

Kevin O'Bryant, A generating function technique for Beatty sequences and other step sequences, Journal of Number Theory, Volume 94, Issue 2, June 2002, Pages 299-319.

Cf. A001113 (e).

Cf. A081544, A081550, A081564.

With[{digmax = 120}, RealDigits[Sum[1/2^Floor[k/E], {k, 1, 20*digmax}], 10, digmax][[1]]] (* Amiram Eldar, May 25 2023 *)

Equals Sum_{k>=1} (1/2)^floor(k/e) = Sum_{k>=1} 1/2^A032634(k).

Data corrected by Amiram Eldar, May 25 2023

A023124 Signature sequence of 1/e (arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

Arrange the numbers i+j*e (i,j >= 1) in increasing order; this sequence is the sequence of j's. - Michel Marcus, Dec 18 2021
If one deletes the first occurrence of 1, the first occurrence of 2, the first occurrence of 3, etc., then the sequence is unchanged. - Brady J. Garvin, Sep 11 2024
Any signature sequence A is closely related to the partial sums of the corresponding homogeneous Beatty sequence: Let Q(d) = d + the sum from g=0 to g=d-1 of floor(theta * g) and Qinv(i) = the maximum integer d such that Q(d) <= i. If there is some d for which Q(d) = i, then A_i = 1. Otherwise, A_i = A_{i - Qinv(i)} + 1. - Brady J. Garvin, Sep 13 2024

J.-P. Delahaye, Des suites fractales d’entiers, Pour la Science, No. 531 January 2022. Sequence h) p. 82.
Clark Kimberling, "Fractal Sequences and Interspersions", Ars Combinatoria, vol. 45 p 157 1997.

T. D. Noe, Table of n, a(n) for n=1..1000
Clark Kimberling, Interspersions
Index entries for sequences related to signature sequences

Cf. A001113, A023123, A032634.

Quiet[Block[{$ContextPath}, Needs["Combinatorica`"]], {General::compat}]
theta = 1 / E;
sums = {0};
cached = <||>;
A023124[i_] := Module[{term, path, base},
  While[sums[[-1]] < i,
    term = sums[[-1]] + Floor[theta * (Length[sums] - 1)] + 1;
    AppendTo[sums, term];
    cached[term] = 1
  ];
  path = {i};
  While[Not[KeyExistsQ[cached, path[[-1]]]],
    AppendTo[path, path[[-1]] - Combinatorica`BinarySearch[sums, path[[-1]]] + 3/2];
  ];
  base = cached[path[[-1]]];
  MapIndexed[(cached[#1] = base + Length[path] - First[#2]) &, path];
  cached[i]
];
Print[Table[A023124[i], {i, 1, 100}]]; (* Brady J. Garvin, Sep 13 2024 *)

from bisect import bisect
from sympy import floor, E
theta = 1 / E
sums = [0]
cached = {}
def A023124(i):
    while sums[-1] < i:
        term = sums[-1] + floor(theta * (len(sums) - 1)) + 1
        sums.append(term)
        cached[term] = 1
    path = [i]
    while path[-1] not in cached:
        path.append(path[-1] - bisect(sums, path[-1]) + 1)
    base = cached[path[-1]]
    for offset, vertex in enumerate(reversed(path)):
        cached[vertex] = base + offset
    return cached[i]
print([A023124(i) for i in range(1, 1001)])  # Brady J. Garvin, Sep 13 2024

A225593 The integer closest to n/e.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26

Offset: 1

José María Grau Ribas, May 11 2013

nonn

Cf. A032634 (with floor).

[Round(n / Exp(1)): n in [1..80]]; // Vincenzo Librandi, May 14 2019

Mathematica
```
Table[Round[n/E], {n, 1, 100}]
```

a(n)=round(n/exp(1)) \\ Charles R Greathouse IV, Nov 13 2013

A062276 a(n) = floor(n^(n+1) / (n+1)^n).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 27, 27, 27, 28, 28

Offset: 0

Henry Bottomley, Jul 02 2001

nonn

a(n) is close to n/e (cf. A032634).

			a(2) = floor(2^3/3^2) = floor(8/9) = 0.

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A032634. Cf. A000169, A007778, A007925.

List([0..90],n->Int(n^(n+1)/(n+1)^n)); # Muniru A Asiru, Jul 01 2018

seq(floor(n^(n+1)/(n+1)^n),n=0..90); # Muniru A Asiru, Jul 01 2018

Array[Floor[#^(# + 1)/(# + 1)^#] &, 78, 0] (* Michael De Vlieger, Jul 01 2018 *)

{ default(realprecision, 50); for (n=0, 1000, write("b062276.txt", n, " ", floor(n^(n + 1) / (n + 1)^n)) ) } \\ Harry J. Smith, Aug 03 2009

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A081573 Decimal expansion of Sum_(1/(2^q-1)) with the summation extending over all pairs of integers gcd(p,q) = 1, 0 < p/q < e = 2.718... .

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A023124 Signature sequence of 1/e (arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x).

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A225593 The integer closest to n/e.

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A062276 a(n) = floor(n^(n+1) / (n+1)^n).

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