A072334 -id:A072334 - cp's OEIS frontend

A338891 a(n) is the least number k such that the average number of odd divisors of {1..k} is >= n.

1, 21, 165, 1274, 9435, 69720, 515230, 3807265, 28132035, 207869515, 1535959665, 11349295155, 83860579775

Offset: 1

Ilya Gutkovskiy, Nov 14 2020

			a(5) = 9435 because the average number of odd divisors of {1..9435} is >= 5.

Eric Weisstein's World of Mathematics, Odd Divisor Function.

Cf. A001227, A060831, A072334, A085829, A328331, A338943.

m = 1; sum = 0; s = {}; Do[sum += DivisorSigma[0, k/2^IntegerExponent[k, 2]]; If[sum >= m*k, AppendTo[s, k]; m++], {k, 1, 10^6}]; s (* Amiram Eldar, Nov 15 2020 *)

a(n) = my(s=1, k=1); while(s>valuation(k, 2))); k; \\ Michel Marcus, Nov 14 2020

a(n+1)/a(n) approaches e^2.

a(11)-a(12) from Amiram Eldar, Nov 16 2020

a(13) from Bill McEachen, Sep 01 2025

A222391 Decimal expansion of e^2/sqrt(Pi).

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Mar 19 2013

			4.1688284832666922304213039077523102603866646811484996378300089546240432...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A072334, A087197, A222392.
Cf. A096789: Sum_{n >= 1} 1/(Gamma(n)*Gamma(n+1)).
Cf. A035009 (see fourth comment).

Digits:=100: evalf(exp(1)^2/sqrt(Pi)); # Wesley Ivan Hurt, Jan 09 2017

Mathematica
```
RealDigits[E^2/Sqrt[Pi], 10, 90][[1]]
```

(exp(1))^2/sqrt(Pi) \\ G. C. Greubel, Jan 09 2017

Equals A072334*A087197.

Equals decimal expansion of Sum_{n >= 1} 1/(Gamma(n/2)*Gamma((n+1)/2)).

A096437 Decimal expansion of (Pi^2-e^2)^(1/2).

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Aug 10 2004

			1.57497565129074563468268199760699

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A002388, A072334, A096255.

RealDigits[Sqrt[Pi^2-E^2],10,120][[1]] (* Harvey P. Dale, Feb 20 2013 *)

sqrt(Pi^2 - exp(2)) \\ G. C. Greubel, Feb 07 2017

A293359 Greatest integer k such that k/2^n < e^2.

Original entry on oeis.org

7, 14, 29, 59, 118, 236, 472, 945, 1891, 3783, 7566, 15132, 30265, 60531, 121062, 242124, 484249, 968498, 1936996, 3873993, 7747986, 15495973, 30991947, 61983895, 123967790, 247935580, 495871160, 991742321, 1983484643, 3966969286, 7933938573, 15867877146

Offset: 0

Clark Kimberling, Oct 10 2017

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A072334, A293360, A293361.

z = 120; r = E^2;
Table[Floor[r*2^n], {n, 0, z}];   (* A293359 *)
Table[Ceiling[r*2^n], {n, 0, z}]; (* A293360 *)
Table[Round[r*2^n], {n, 0, z}];   (* A293361 *)

a(n) = floor(r*2^n), where r = e^2.

a(n) = A293360(n) - 1.

A293360 Least integer k such that k/2^n > e^2.

Original entry on oeis.org

8, 15, 30, 60, 119, 237, 473, 946, 1892, 3784, 7567, 15133, 30266, 60532, 121063, 242125, 484250, 968499, 1936997, 3873994, 7747987, 15495974, 30991948, 61983896, 123967791, 247935581, 495871161, 991742322, 1983484644, 3966969287, 7933938574, 15867877147

Offset: 0

Clark Kimberling, Oct 11 2017

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A072334, A293359, A293361.

z = 120; r = E^2;
Table[Floor[r*2^n], {n, 0, z}];   (* A293359 *)
Table[Ceiling[r*2^n], {n, 0, z}]; (* A293360 *)
Table[Round[r*2^n], {n, 0, z}];   (* A293361 *)

a(n) = ceiling(r*2^n), where r = e^2.

a(n) = A293359(n) + 1.

A293361 The integer k that minimizes |k/2^n - e^2|.

Original entry on oeis.org

7, 15, 30, 59, 118, 236, 473, 946, 1892, 3783, 7566, 15133, 30266, 60531, 121062, 242125, 484249, 968498, 1936997, 3873993, 7747987, 15495974, 30991948, 61983895, 123967790, 247935580, 495871161, 991742322, 1983484643, 3966969287, 7933938573, 15867877147

Offset: 0

Clark Kimberling, Oct 11 2017

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A072334, A293359, A293360.

z = 120; r = E^2;
Table[Floor[r*2^n], {n, 0, z}];   (* A293359 *)
Table[Ceiling[r*2^n], {n, 0, z}]; (* A293360 *)
Table[Round[r*2^n], {n, 0, z}];   (* A293361 *)

a(n) = floor(1/2 + (e^2)*2^n).

a(n) = A293359(n) if (fractional part of (e^2)*2^n) < 1/2, else a(n) = A293360(n).

A368654 Decimal expansion of 158452/21444.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Jan 02 2024

It is a rational approximation of e^2 (A072334) provided by Charles Hermite in 1874 (see Hermite and Maor).

Periodic with a period length of 893. - Ray Chandler, Jan 19 2024

			7.38910650997948144002984517813840701361686...

Eli Maor, e: The Story of a Number. Princeton, New Jersey: Princeton University Press (1994), p. 189.

Philip R. Brown, Approximations of e and Pi: an exploration, International Journal of Mathematical Education in Science and Technology, 48:sup1, S30-S39, 2017. See p. S32.
Charles Hermite, Sur la fonction exponentielle, Gauthier-Villars, Paris, 1874.
Index entries for sequences related to the number e
Index entries for linear recurrences with constant coefficients, order 893.

Cf. A001113, A007676, A007677, A072334, A368617, A368653, A368656.

Flatten[First[RealDigits[158452/21444,10,100]]]

A067840 Factorial expansion of e^2 : exp(2) = Sum_{n >= 0} a(n)/n!.

Original entry on oeis.org

7, 0, 0, 2, 1, 1, 4, 0, 6, 6, 6, 8, 5, 11, 5, 10, 10, 15, 16, 8, 19, 18, 15, 0, 16, 1, 2, 26, 17, 27, 23, 17, 18, 11, 24, 34, 25, 16, 27, 5, 33, 20, 11, 39, 35, 25, 39, 7, 5, 21, 27, 30, 33, 21, 34, 9, 10, 26, 32, 15, 35, 23, 6, 3, 21, 43, 50, 40, 41, 33, 1, 62, 58, 59, 12, 23, 62, 42

Offset: 0

Benoit Cloitre, Mar 10 2002

Cf. A072334, A007514.

a(0) = 7; for n>=1, a(n) = floor(n!*e^2) - n*floor((n-1)!*e^2).

Offset changed to 0 by Sean A. Irvine, Jan 09 2024

A182767 Beatty sequence for 1+e^2.

Original entry on oeis.org

8, 16, 25, 33, 41, 50, 58, 67, 75, 83, 92, 100, 109, 117, 125, 134, 142, 151, 159, 167, 176, 184, 192, 201, 209, 218, 226, 234, 243, 251, 260, 268, 276, 285, 293, 302, 310, 318, 327, 335, 343, 352, 360, 369, 377, 385, 394, 402, 411, 419, 427

Offset: 1

Clark Kimberling, Nov 29 2010

nonn

Let u=e=A001113 and v=1/e=A068985. Jointly rank {j*u} and {k*v} as in the first comment at A182760; a(n) is the position of n*u.

Cf. A182760, A182768.

A182767 := proc(n) floor(n*(1+exp(2))) ; end proc:

a(n)=floor(n*(1+e^2)) = floor(n+n*A072334).

A225141 Primes from merging of 10 successive digits in the decimal expansion of exp(2).

Original entry on oeis.org

7460575007, 2252257379, 6079057763, 5316126547, 7447839221, 5414146799, 2933188807, 8997407299, 7407299869, 2998696009, 9869600953, 2368469479, 6947930299, 7714456831, 1445683123, 6619147987, 8715043283, 7724849201, 8492011193, 1193531621, 2117195173

Offset: 1

Bruno Berselli, Apr 30 2013

Leading zeros are not permitted, so each prime is 10 digits in length. The terms are listed in the order in which they occur.

Bruno Berselli, Table of n, a(n) for n = 1..1000
Simon Plouffe, exp(2) to 5000 digits

Cf. A072334, A104998 - A105005.

Cf. A105006.

With[{len = 10}, FromDigits /@ Select[Partition[RealDigits[E^2, 10, 700][[1]], len, 1], PrimeQ[FromDigits[#]] && IntegerLength[FromDigits[#]] == len &]]

cp's OEIS Frontend

A338891 a(n) is the least number k such that the average number of odd divisors of {1..k} is >= n.

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A222391 Decimal expansion of e^2/sqrt(Pi).

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A096437 Decimal expansion of (Pi^2-e^2)^(1/2).

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A293359 Greatest integer k such that k/2^n < e^2.

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A293360 Least integer k such that k/2^n > e^2.

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A293361 The integer k that minimizes |k/2^n - e^2|.

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A368654 Decimal expansion of 158452/21444.

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A067840 Factorial expansion of e^2 : exp(2) = Sum_{n >= 0} a(n)/n!.

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