Marvin Ray Burns - cp's OEIS frontend

A248660 Simple continued fraction expansion of the constant defined in A037077.

0, 5, 3, 10, 1, 1, 4, 1, 1, 1, 1, 9, 1, 1, 12, 2, 17, 2, 2, 1, 1, 17, 1, 6, 4, 1, 3, 3, 4, 2, 1, 262, 2, 1, 4, 1, 49, 2, 1, 9, 1, 2, 1, 1, 4, 23, 26, 6, 6, 5, 3, 3, 1, 1, 1, 144, 9, 1, 1, 5, 1, 3, 1, 1, 5, 13, 8619, 2, 1, 45, 2, 1, 1, 2, 1, 4, 5, 1, 7, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 6, 3, 1, 1, 2, 2, 7, 3, 136, 1

Offset: 0

Marvin Ray Burns, Jan 11 2015

A037077 is sometimes called the MRB constant.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 450.

Eric Weisstein's World of Mathematics, MRB Constant

Cf. A037077 (decimal expansion).

MRB:= Sum((-1)^k*(k^(1/k)-1),k=1..infinity):
V:= evalf[150](MRB):
subs(`...`=NULL, numtheory:-cfrac(V, 100, 'quotients')); # Robert Israel, Jan 12 2015

m = NSum[(-1)^n*(n^(1/n) - 1), {n, 1, Infinity},
   WorkingPrecision -> 100, Method -> "AlternatingSigns"];
ContinuedFraction[m]

contfrac(sumalt(x=1, (-1)^x*((x^(1/x))-1)) ) \\ Michel Marcus, Jan 12 2015

A231336 Integers n such that appending some decimal digit to the first n digits of Pi results in a prime.

Original entry on oeis.org

0, 1, 2, 5, 11, 12, 18, 37, 39, 77, 82, 100, 125, 128, 220, 305, 601, 676, 1692, 1901, 2202, 2253, 2394, 3318, 3970, 5826, 7001, 9853, 12607, 13434, 16207

Offset: 1

Marvin Ray Burns and Robert G. Wilson v, Nov 07 2013

A140515 is a proper subsequence. A060421 - 1 is a proper subsequence. So the terms 47576 & 78072 are also members.

			0 is in the sequence since 2, 3, 5, and 7 are all primes;
1 is in the sequence since 31 and 37 are both primes;
2 is in the sequence since 311, 313, and 317 are all primes;
3 is not in the sequence since 3141, 3143, 3147, and 3149 are all composites;
4 is not in the sequence since 31411, 31413, 31417, and 31419 are all composites;
5 is in the sequence since 314159 is a prime; etc.

Cf. A140515, A060421, A005042.

fQ[n_] := Union[PrimeQ[ 10 IntegerPart[10^n*Pi] + {1, 3, 7, 9}]][[-1]]; k = -1; lst = {}; While[k < 17001, If[ fQ@ k, AppendTo[lst, k + 1]; Print[k + 1]]; k++]; lst
Module[{nn=16300,pd},pd=RealDigits[Pi,10,nn][[1]];Select[Range[0,nn],AnyTrue[ 10*FromDigits[Take[pd,#]]+{1,3,7,9},PrimeQ]&]] (* Harvey P. Dale, Aug 14 2022 *)

is(n)=my(d=Pi*10^n\10*10);isprime(d+1) || isprime(d+3) || isprime(d+7) || isprime(d+9) \\ Charles R Greathouse IV, Nov 07 2013

Keyword "base" added by Zak Seidov, Nov 11 2013

A214128 a(n) = 6^(6^6) mod n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 0, 6, 5, 0, 1, 8, 6, 0, 1, 0, 1, 16, 15, 16, 2, 0, 6, 14, 0, 8, 23, 6, 1, 0, 27, 18, 1, 0, 1, 20, 27, 16, 18, 36, 1, 16, 36, 2, 37, 0, 43, 6, 18, 40, 44, 0, 16, 8, 39, 52, 5, 36, 9, 32, 36, 0, 1, 60, 14, 52, 48, 36, 6, 0, 1, 38, 6, 20, 71

Offset: 1

Marvin Ray Burns, Jul 04 2012

nonn

The indices of zeros in this sequence, i.e., divisors of 6^(6^6), are all numbers of the form 2^i * 3^j, with 0 <= i, j <= 6^6. [Edited by M. F. Hasler, Feb 25 2018]
If c and N are any positive integers, and p^k is the largest prime power divisor of c, then the divisors of c^N less than p^(k*N+1) are precisely those numbers in that range whose prime factorization includes only primes that divide c. This is the case c = 6, N = 6^6, so p^k = 2^1 = 2; so the first difference in the divisor list from A003586 is for A003586(n) = 2^(6^6+1). Franklin T. Adams-Watters, Jul 12 2012
Eventually constant: see formula. - M. F. Hasler, Feb 24 2018
If n > 1 is coprime to 6 and A000010(n) divides 6^6 then a(n)=1. - Robert Israel, Nov 27 2019

			a(1) = 6^(6^6) mod 1 = 0.
a(2) = 6^(6^6) mod 2 = 0.
a(3) = 6^(6^6) mod 3 = 0.
a(4) = 6^(6^6) mod 4 = 0.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for eventually constant sequences.

Cf. A129810 (9^9^9 mod n), A003586.

seq(6 &^ (6^6) mod n, n=1..100); # Robert Israel, Nov 27 2019

Mathematica
```
Table[PowerMod[6, 6^6, n], {n, 100}]
```

a(n)=lift(Mod(6,n)^6^6) \\ Charles R Greathouse IV, Jul 05 2012

a(n) = 0 if and only if n = 2^i 3^j, 0 <= i, j <= 6^6; after the last of these zeros at n = 6^6^6, a(n) = 6^6^6 for all n > 6^6^6 ~ 2.659*10^36305. - M. F. Hasler, Feb 24 2018

A177218 Decimal expansion of the integral over cos(Pix)x^(1/x) between 1/e and e.

Original entry on oeis.org

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

Offset: 0

Marvin Ray Burns, May 04 2010

Strangely close to A037077 which is a sum of the integrand from 1 to infinity.

			0.187779...

Marvin Ray Burns, Author's original inquiry
R. J. Mathar, Numerical evaluation of the oscillatory integral over exp(i*pi*x)x^(1/x) between 1 and infinity, arXiv:0912.3844

A157852 is the same integral from 1 to infinity.

Int( cos(Pi*x)*x^(1/x),x=exp(-1)..exp(1)) ; evalf(%) ; # R. J. Mathar, May 07 2010

RealDigits[ Re[NIntegrate[(-1)^n*n^(1/n), {n, 1/E, E}, WorkingPrecision -> 200]]]

Definition simplified, keyword:cons inserted, offset corrected by R. J. Mathar, May 07 2010

A173273 Decimal expansion of 1 - 2*A037077.

Original entry on oeis.org

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

Offset: 0

Marvin Ray Burns, Feb 14 2010, Feb 24 2010, Mar 05 2010

			0.624280715...

Cf. A037077.

digits = 105; 1-2*NSum[ (-1)^n*((n^(1/n)) - 1), {n, 1, Infinity}, WorkingPrecision -> digits+10, Method -> "AlternatingSigns"] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 15 2013 *)

A157852 Decimal expansion of the absolute value of lim_{N -> infinity} Integral_{x=1..2N} e^(iPix)x^(1/x).

Original entry on oeis.org

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

Offset: 0

Marvin Ray Burns, Mar 07 2009

This constant is the integral analog of the constant described in A037077 since e^(i*Pi*x) =(-1)^x. While A037077 was named the MRB constant by Simon Plouffe, Marvin Ray Burns named this constant MKB after his wife at the time.
This constant is hard to integrate and very slow to converge, so it takes a combination of modern methods to calculate many digits!
This constant could be written as a special value, for omega=Pi, of the function f(omega) = lim_{N->infinity} Integral_{x = Pi/omega .. 2N8(Pi/omega)} (exp(i*omega*x)*x^(1/x)), a kind of discretely sampled Fourier transform of x^(1/x). This stresses the fact that it is a complex entity. People who desire to underline the similarity of this integral to the MRB alternating series (A037077) often write the factor exp(i*Pi*x) as (-1)^x, which can be a bit confusing because it hides the imaginary unit. - Stanislav Sykora, Apr 08 2016

			After integrating from 1 to 15 million the absolute value of the integral is approximately 0.687652_7, after integrating from 1 to 20 million approximately 0.687652_6.

Marvin Ray Burns, Table of n, a(n) for n = 0..19999
Marvin Ray Burns, Author's public inquiry 1
Marvin Ray Burns, Author's public inquiry 2
Marvin Ray Burns, How I calculated the digits of the MKB constant
Marvin Ray Burns, Paper on 20000 digits in a Mathematica notebook (Digits checked by different formula, computing more digits)
Marvin Ray Burns, How 40,000 digits of A157852 was successfully computed and checked to 35,000 digits
Marvin Ray Burns, Mathematica Notebook of 54,386 digits of A157852 computed with increasing MaxRecursion, with proof at bottom.
Marvin Ray Burns, How_I_found_A157852_sum.
Marvin Ray Burns, A157852 sum PDFK
R. J. Mathar, Numerical evaluation of the oscillatory integral over exp(i*pi*x)x^(1/x) between 1 and infinity, arxiv:0912.3844 [math.CA], 2009-2010.

Integrating A037077 instead of summing.

Cf. A037077, A255727 (real part), A255728 (imaginary part).

# After Marvin Ray Burns's "Program 3".
f := (n, x) -> seq(x, 0..n):
m := n -> (Pi/I)^n * MeijerG([[], [f(n, 1)]], [[1-n, f(n, 0)], []], -I*Pi):
s := n -> abs(add(m(k), k = 1..n) - 2)/Pi:
# s(n) approximates the constant for n -> oo and suitable chosen precision.
seq(evalf(s(n), 22), n = 1..3); # Peter Luschny, Nov 16 2021

a = NIntegrate[ x^(1/x)*Cos[Pi*x], {x, 1, 10^20}, WorkingPrecision -> 30, MaxRecursion -> 70]; b = NIntegrate[ x^(1/x)*Sin[Pi*x], {x, 1, 10^20}, WorkingPrecision -> 30, MaxRecursion -> 70]; RealDigits[ Sqrt[a^2 + b^2], 10, 18] // First (* Jean-François Alcover, Feb 14 2013 *)
(* Program 2: to compute and verify 1000s of digits through a different formula. *)
g[x_] = x^(1/x); t = (Timing[
    MKB = -(I NIntegrate[(g[(1 + t I)]) ( Exp[-Pi t]), {t, 0,
           Infinity}, WorkingPrecision -> 2410,
          Method -> "Trapezoidal", MaxRecursion -> 10] + I/Pi)])[[
  1]]; Print["Timing for calculation=", t]; t = (Timing[
    MKB1 = (1/Pi  NIntegrate[
         g'[1 + I t] Exp[-Pi t], {t, 0, Infinity},
         WorkingPrecision -> 2410, Method -> "Trapezoidal",
         MaxRecursion -> 10] - 2 I/Pi)])[[
  1]]; Print["Timing for verification=", t]; err =
test - test2; Print["Error=", N[err, 20]];Abs[MKB] (* MaxRecursion -> 13 works for 10,000 digits. Marvin Ray Burns, Apr 18 2021 *)
(* Program 3: An infinite sum involving the Meijer G function. Compare the discussion near the end of "How I calculated the digits of the MKB constant" and all of Cloud Notebook "How_I_found_A157852_sum" in the link section. *)
f[n_] := MeijerG[{{},Table[1, {n+1}]}, {Prepend[Table[0, n+1], -n + 1], {}}, -I Pi];
Abs[Sum[(I/Pi)^(1 - n) N[f[n], 22], {n, 1, 15}] - 2 I/Pi] (* Marvin Ray Burns, Nov 15 2021 *)

Equals sqrt(A255727^2 + A255728^2). - Joerg Arndt, Apr 05 2016

Edited by N. J. A. Sloane, Mar 13 2009
Corrected and edited by Marvin Ray Burns, Apr 03 2009
8 more digits from R. J. Mathar, Nov 30 2009, 3 more Jan 03 2011, 3 more on Feb 25 2013
15 more digits added by Marvin Ray Burns, Feb 26 2013
Many more digits added by Marvin Ray Burns, May 11 2015
Edited by Marvin Ray Burns, Aug 06 2015
Edited by Marvin Ray Burns, Jun 18 2017

A160755 Number of correct digits of the MRB constant derived from the sequence of partial sums up to m=10^n terms as defined by S[n]= Sum[(-1)^k*(k^(1/k)-1),{k,m}].

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49

Offset: 1

Marvin Ray Burns, May 25 2009

Adding the series -1+sqrt(2)-3^(1/3)+4^(1/4)..., according to this sequence, 10 billion terms must be added to arrive at 11 accurate digits of the MRB constant.

			For n=1, a(n)=1 because after 10^1 partial sums of -1+sqrt(2)-3^(1/3)+4^(1/4)... you get one accurate digit of the MRB constant.
For n=2, a(n)=2 because after 10^2 partial sums you get two accurate digits and so on.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 450. ISBN 0521818052.

Henri Cohen, Fernando Rodriguez Villegas and Don Zagier, Convergence Acceleration of Alternating Series, Experimental Mathematics, 9:1 (2000).
Eric Weisstein's World of Mathematics, MRB Constant.
Wikipedia, Mathematical constant

Cf. A037077 (the MRB constant).

m = NSum[(-1)^n*(n^(1/n) - 1), {n, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 1000]; Table[-Floor[Log[10, Abs[m - NSum[(-1)^n*(n^(1/n) - 1), {n, 10^a}, Method ->"AlternatingSigns", WorkingPrecision -> 1000]]]], {a,1, 50}]

Corrections from Marvin Ray Burns, Jun 05 2009
Link to Wikipedia replaced by up-to-date version; keyword:less added R. J. Mathar, Aug 04 2010
Corrections by Marvin Ray Burns, Aug 21 2010, Jul 15 2012

A140515 Numbers n such that one of floor(10^n * Pi) or ceiling(10^n * Pi) is prime.

Original entry on oeis.org

0, 1, 5, 11, 18, 37, 601, 1901, 2394, 3970, 5826, 16207

Offset: 1

Marvin Ray Burns, Jul 01 2008, Jul 02 2008

			ceiling(10^11*Pi) = ceiling(314159265358.9...) = 314159265359 is prime, so 11 is in the sequence.

Carlos B. Rivera F. Puzzle 50. Approximation to pi with primes, The Prime Puzzles and Problems Connection.
Eric Weisstein's World of Mathematics, Pi Digits

npQ[n_]:=Module[{c=10^n Pi},Total[Boole[PrimeQ[{Floor[ c],Ceiling[ c]}]]] == 1]; Select[Range[0,4000],npQ] (* The program generates the first 10 terms of the sequence. To generate more, increase the Range constant, but the program may take a long time to run. *) (* Harvey P. Dale, May 07 2021 *)

isA140515(n)=isprime(bitor(floor(10^n*Pi),1))

A135373 a(n) = prime(2^(n + 1)) - prime(2^n).

Original entry on oeis.org

1, 4, 12, 34, 78, 180, 408, 900, 2052, 4490, 9702, 21010, 45144, 96486, 205590, 435548, 920896, 1938594, 4072946, 8535970, 17845982, 37242540, 77569570, 161300102, 334906956, 694329000, 1437620932, 2972973918, 6141189354, 12672042908, 26122458858, 53799031890

Offset: 0

Marvin Ray Burns, Dec 09 2007

nonn

Cf. A000040, A033844, A135371.

Table[Prime[2^(n + 1)] - Prime[2^n], {n, 0, 40}]
#[[2]]-#[[1]]&/@Partition[Prime[2^Range[0,30]],2,1] (* Harvey P. Dale, Feb 21 2015 *)

a(n) = prime(2^(n + 1)) - prime(2^n); \\ Michel Marcus, Nov 15 2020

a(n) = A033844(n+1) - A033844(n). - Jinyuan Wang, Nov 14 2020

a(29)-a(31) from Jinyuan Wang, Nov 14 2020

A135371 a(n) = prime(3^(n + 1)) - prime(3^n).

Original entry on oeis.org

3, 18, 80, 316, 1124, 3976, 13770, 46398, 155174, 512700, 1679218, 5454768, 17617664, 56578816, 180896064, 576041630, 1827788446, 5781493590, 18236550884, 57379347420, 180130119286, 564319058202, 1764618506934, 5508483994106, 17168384214168, 53431296170784

Offset: 0

Marvin Ray Burns, Dec 09 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 0..35 (calculated from the b-file at A038833)

Cf. A000040, A038833, A110896, A135373.

Table[Prime[3^(n + 1)] - Prime[3^n], {n, 0, 26}]

a(n) = prime(3^(n + 1)) - prime(3^n); \\ Michel Marcus, Nov 15 2020

a(n) = A038833(n+1) - A038833(n). - Jinyuan Wang, Nov 14 2020

a(23)-a(25) from Jinyuan Wang, Nov 14 2020

cp's OEIS Frontend

User: Marvin Ray Burns

A248660 Simple continued fraction expansion of the constant defined in A037077.

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A231336 Integers n such that appending some decimal digit to the first n digits of Pi results in a prime.

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A214128 a(n) = 6^(6^6) mod n.

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A177218 Decimal expansion of the integral over cos(Pi*x)*x^(1/x) between 1/e and e.

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A173273 Decimal expansion of 1 - 2*A037077.

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A157852 Decimal expansion of the absolute value of lim_{N -> infinity} Integral_{x=1..2*N} e^(i*Pi*x)*x^(1/x).

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A160755 Number of correct digits of the MRB constant derived from the sequence of partial sums up to m=10^n terms as defined by S[n]= Sum[(-1)^k*(k^(1/k)-1),{k,m}].

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A177218 Decimal expansion of the integral over cos(Pix)x^(1/x) between 1/e and e.

A157852 Decimal expansion of the absolute value of lim_{N -> infinity} Integral_{x=1..2N} e^(iPix)x^(1/x).