Stefan Steinerberger - cp's OEIS frontend

A347520 A053392 with duplicates removed.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 210, 32, 33, 34, 35, 36, 37, 38, 39, 310, 311, 43, 44, 45, 46, 47, 48, 49, 410, 411, 412, 54, 55, 56, 57, 58, 59

Offset: 0

Stefan Steinerberger, Sep 04 2021

nonn

Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], 2020.

Cf. A053392.

a[n_] := Total /@ Transpose[{Most[id = IntegerDigits[n]], Rest[id]}] //
     IntegerDigits // Flatten // FromDigits; DeleteDuplicates[Table[a[n], {n, 0, 1000}]]

# uses A053392
from collections import OrderedDict
def afiltern(terms):
    return list(OrderedDict.fromkeys(A053392(k) for k in range(terms)))
print(afiltern(179)) # Michael S. Branicky, Sep 04 2021

A114575 Number of distinct prime factors of floor(e^n).

Original entry on oeis.org

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

Offset: 0

Stefan Steinerberger, Feb 17 2006

nonn

			floor(e^3) = floor(20.08553) = 20. 20 has two distinct prime factors (2 and 5), therefore a(3) = 2.

Cf. A001113 [decimal expansion of e].

Table[Length[FactorInteger[Floor[E^n]]], {n, 1, 80}]
PrimeNu[Floor[E^Range[80]]] (* Harvey P. Dale, Jul 21 2013 *)

A114427 Decimal expansion of the real solution of x^3-x^2-x-4=0.

Original entry on oeis.org

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

Offset: 1

Stefan Steinerberger, Feb 13 2006

The solution of the equation is twice the value of a lower bound on randomly generated Fibonacci-like sequences.

			The solution of the equation is 2.2418965630344798557894...

Eran Makover and Jeffrey McGowan, An elementary proof that random Fibonacci sequences grow exponentially, Journal of Number Theory, Volume 121, Issue 1, November 2006, Pages 40-44. (arXiv:math/0510159 [math.NT])

Cf. A114431.

RealDigits[1/3 (1+(1/2 (119-3 Sqrt[1545]))^(1/3)+(1/2 (119+3 Sqrt[ 1545]))^(1/3)),10,120][[1]] (* Harvey P. Dale, Jun 09 2011 *)

A114594 Decimal expansion of (e^Pi)*(Pi^e).

Original entry on oeis.org

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

Offset: 3

Stefan Steinerberger, Feb 21 2006

			(e^Pi)*(Pi^e) is 519.7204655517051016...

Cf. A019314 (sum of e^Pi and Pi^e).

a := N[E^Pi*Pi^E, 500]; Table[Floor[10^(n - 3)*a] - 10*Floor[10^(n - 4)*a], {n, 1, 100}] (* valid for the first 500 digits *)
RealDigits[N[Pi^E*E^Pi,200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

A117672 Numbers n such that |cos(n)*cos(n+2)| < (cos(n+1))^2.

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 21, 24, 25, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 43, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 65, 68, 69, 71, 72, 74, 75, 77, 78, 80, 81, 83, 84, 87, 90, 91, 93, 94, 96, 97, 99, 100, 102, 103, 105, 106, 109, 112, 113, 115

Offset: 1

Stefan Steinerberger, Apr 12 2006

nonn

What is the density of the sequence? I am convinced that this sequence has the same density as A026313.

			|cos(5)*cos(7)| < (cos(6))^2, therefore 5 is in the sequence.

Cf. A026313 [same definition with sin instead of cos].

Select[Range[120], Abs[Cos[ # ]*Cos[ # + 2]] < Cos[ # + 1]^2 &]

Offset corrected by Sean A. Irvine, Sep 25 2019

A114573 Numbers k such that phi(k) is a perfect 11th power.

Original entry on oeis.org

1, 2, 3855, 4096, 4112, 4352, 5120, 5140, 5440, 6144, 6168, 6528, 7680, 7710, 8160, 5570645, 8388608, 8388736, 8421376, 8912896, 8913032, 8947712, 10485760, 10485920, 10526720, 11141120, 11141290, 11184640, 12582912, 12583104

Offset: 1

Stefan Steinerberger, Feb 17 2006

nonn

Given the fact that phi(n) > sqrt(n) for all n except n=2 and n=6 we can see that every 11th power does appear as value only a finite number of times. What bounds on the density of this sequence can be proved?

			phi(4096) = 2048 = 2^11.

Amiram Eldar, Table of n, a(n) for n = 1..3625

Cf. A039770 (square), A039771 (cube), A078164 (4th), A078165 (5th), A078166 (6th), A078167 (7th), A078168 (8th), A078169 (9th), A078170 (10th power), A000010.

For[n = 1, n < 100000, n++, If[EulerPhi[n]^(1/11) == Floor[EulerPhi[n]^(1/11)], Print[n]]]

More terms from Stefan Steinerberger, May 16 2007

A114431 Decimal expansion of the real solution of x^3 - x^2 - 2x - 4 = 0.

Original entry on oeis.org

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

Offset: 0

Stefan Steinerberger, Feb 13 2006

The solution of the equation is twice the value of an upper bound on randomly generated Fibonacci-like sequences.

Also, 1/log_2(x), where x is this constant, is the exponent in the exponent of the growth rate of the first Grigorchuk group. - Andrey Zabolotskiy, Apr 14 2020

Anna Erschler & Tianyi Zheng, Growth of periodic Grigorchuk groups, Invent. math. 219, 1069-1155 (2020).
Eran Makover and Jeffrey McGowan, An elementary proof that random Fibonacci sequences grow exponentially, Journal of Number Theory, Volume 121, Issue 1, November 2006, Pages 40-44. (arXiv:math/0510159 [math.NT])
Wikipedia, Grigorchuk group

Cf. A114427.

RealDigits[x/.FindRoot[x^3 - x^2 - 2x == 4, {x, 2}, WorkingPrecision -> 120], 10, 120] [[1]] (* or *) RealDigits[(1 + Surd[64 - 3 * Sqrt[417], 3] + Surd[64 + 3 * Sqrt[417], 3])/3, 10, 120][[1]] (* Harvey P. Dale, Dec 02 2017 *)

default(realprecision, 105); 1/3*(1+(64-3*sqrt(417))^(1/3)+(64+3*sqrt(417))^(1/3)) \\ Michel Marcus, Jun 14 2013

A114941 Decimal expansion of the infinite sum Sum_{k>=1} cos(k)/k!.

Original entry on oeis.org

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

Offset: 0

Stefan Steinerberger, Feb 21 2006

			0.143835643791640325906...

Cf. A001113, A049469, A049470, A114940.

RealDigits[Sum[N[Cos[i],400]/i!, {i,1,300}]][[1]] (*which is accurate to 300 digits*) (* corrected by Harvey P. Dale, Nov 29 2011 *)

suminf(k=1, cos(k)/k!) \\ Michel Marcus, Jul 19 2020

From Amiram Eldar, Jul 19 2020: (Start)
Equals e^cos(1) * cos(sin(1)) - 1.
Equals cos(sin(1)) * (cosh(cos(1)) + sinh(cos(1))) - 1.
Equals (e^(e^i) + e^(e^(-i)))/2 - 1. (End)

A115566 Numbers k such that 2^k, 2^(k+1) and 2^(k+2) have the same number of digits.

Original entry on oeis.org

1, 4, 7, 10, 11, 14, 17, 20, 21, 24, 27, 30, 31, 34, 37, 40, 41, 44, 47, 50, 51, 54, 57, 60, 61, 64, 67, 70, 71, 74, 77, 80, 81, 84, 87, 90, 91, 94, 97, 100, 103, 104, 107, 110, 113, 114, 117, 120, 123, 124, 127, 130, 133, 134, 137, 140, 143, 144, 147, 150, 153, 154

Offset: 1

Stefan Steinerberger, Mar 11 2006

The density of this sequence is 1 - 2*log_10(2) = 0.3979400086720376...

			2^4 = 16, 2^5 = 32, 2^6 = 64: all these numbers have two digits.
2^10 = 1024, 2^11 = 2048, 2^12 = 4096: all these numbers have three digits.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001682 (same definition with 3 instead of 2).

Cf. A034887 (number of digits in 2^n).

[k:k in [1..160]|#Intseq(2^k) eq #Intseq(2^(k+2))]; // Marius A. Burtea, May 20 2019

select(n -> ilog10(2^n)=ilog10(2^(n+2)), [$1..1000]); # Robert Israel, May 19 2019

Select[Range[220], Floor[Log[10, 2]*# ] == Floor[Log[10, 2]*(# + 2)] &]

floor(log_10(2)*k) = floor(log_10(2)*(k+1)) = floor(log_10(2)*(k+2)).

A113887 Numbers n such that floor(exp(sqrt(n))) is a prime number.

Original entry on oeis.org

1, 3, 4, 6, 10, 12, 17, 21, 26, 30, 53, 54, 58, 83, 95, 109, 111, 128, 131, 137, 145, 157, 165, 166, 181, 195, 202, 244, 261, 265, 290, 306, 324, 343, 353, 369, 386, 415, 417, 418, 438, 468, 473, 503, 633, 704, 735, 758, 859, 903, 919, 955, 979, 987, 1008, 1016

Offset: 1

Stefan Steinerberger, Jan 28 2006

nonn

Cf. A050808.

Select[ Range@1044, PrimeQ@ Floor@ Exp@ Sqrt@ # &] (* Robert G. Wilson v *)

More terms from Robert G. Wilson v, Jan 30 2006

cp's OEIS Frontend

User: Stefan Steinerberger

A347520 A053392 with duplicates removed.

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A114575 Number of distinct prime factors of floor(e^n).

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A114427 Decimal expansion of the real solution of x^3-x^2-x-4=0.

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A114594 Decimal expansion of (e^Pi)*(Pi^e).

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A117672 Numbers n such that |cos(n)*cos(n+2)| < (cos(n+1))^2.

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A114573 Numbers k such that phi(k) is a perfect 11th power.

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A114431 Decimal expansion of the real solution of x^3 - x^2 - 2x - 4 = 0.

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A114941 Decimal expansion of the infinite sum Sum_{k>=1} cos(k)/k!.

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A115566 Numbers k such that 2^k, 2^(k+1) and 2^(k+2) have the same number of digits.

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