A068985 -id:A068985 - cp's OEIS frontend

A072364 Decimal expansion of (1/e)^(1/e).

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

Offset: 0

Rick L. Shepherd, Jul 18 2002

Minimum value of x^x for real x>0.
Also minimum value of 1/x^(1/x) for real x>0 (occurs at e). Equals exp(Pi)/exp(1/exp(1)) * exp(-Pi). - Gerald McGarvey, Sep 21 2004
If (1/e)^(1/e) < y < 1, then x^x = y has two solutions x = a and x = b with 0 < a < 1/e < b < 1. For example, (1/e)^(1/e) < 1/sqrt(2) < 1 and (1/4)^(1/4) = (1/2)^(1/2) = 1/sqrt(2) with 1/4 < 1/e < 1/2. - Jonathan Sondow, Sep 02 2011

			0.69220062755534635386...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 26, page 233.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Alex Chin et al., Pick a Tree-Any Tree, The American Mathematical Monthly, 122.5 (2015): 424-432.
Plouffe's Inverter entry for .69220062755.
Jonathan Sondow and Diego Marques, Algebraic and transcendental solutions of some exponential equations, arXiv:1108.6096 [math.NT], 2011; Annales Mathematicae et Informaticae 37 (2010) 151-164; see p. 3 in the link.

Cf. A068985 (1/e), A001113 (e), A072365 ((1/3)^(1/3)), A073229 (e^(1/e)), A073230 ((1/e)^e).

Cf. also A258707.

(Exp(-1))^(Exp(-1)); // G. C. Greubel, May 29 2018

evalf(exp(-1/exp(1)), 120);  # Alois P. Heinz, Oct 26 2021

Mathematica
```
RealDigits[E^(-1/E), 10, 111][[1]]
```
PARI
```
(1/exp(1))^(1/exp(1))
```

exp(-1/exp(1)) \\ Charles R Greathouse IV, Sep 01 2011

From Amiram Eldar, Aug 19 2020: (Start)
Equals Sum_{k>=0} (-1)^k/(exp(k)*k!).
Equals Product_{k>=0} exp((-1)^(k+1)/k!). (End)

A092553 Decimal expansion of 1/e^2.

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 09 2004

Consider a substrate (such as polyvinyl alcohol or in forming the polymer of methyl vinyl ketone) in a "1,3 configuration" in which substituents branching off the substrate can irreversibly join with neighboring substituents unless the neighbor is already joined to its other neighbor. Then this constant is the fraction of joined substituents on an infinite substrate.
This also applies to reversible reactions when the rate of forward reaction is much faster than that of backward reaction; see Flory p. 1518 footnote 5. This had "satisfactory accord" with his experimental data using methyl vinyl ketone polymer for which the experimentally-obtained percentage was 0.15.
(A 1,k configuration is a substituent branching off a carbon atom, k-2 intermediate carbon atoms, and substituent branching off a carbon atom.)  - Charles R Greathouse IV, Nov 30 2012
Also the probability, as n increases without bound, that a permutation of length n is simple: no intervals of length 1 < k < n (an interval of a permutation s is a set of contiguous numbers which in s have consecutive indices). - Charles R Greathouse IV, May 14 2014

			0.1353352832366...

M. H. Albert, M. D. Atkinson and M. Klazar, The enumeration of simple permutations, J. Integer Seq. 6 (2003) 03.4.4. arXiv:math/0304213.
R. Brignall, A survey of simple permutations, Permutation Patterns, ed. S. Linton, N. Ruškuc and V. Vatter, Cambridge Univ. Press, 2010, pp. 41—65; arXiv:0801.0963.
Paul J. Flory, Intramolecular reaction between neighboring substituents of vinyl polymers, Journal of the American Chemical Society 61:6 (1939), pp. 1518-1521.
Index entries for transcendental numbers

Cf. A019774, A001113, A068985, A219863.

RealDigits[N[1/E^2,200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

exp(-2) \\ Charles R Greathouse IV, Nov 30 2012

From Peter Bala, Oct 27 2019: (Start)
1/e^2 = Sum_{k >= 0} (-2)^k/k!.
This is the case n = 0 of the following series acceleration formulas:
1/e^2 = n!*2^n*Sum_{k >= 0} (-2)^k/(k!*R(n,k)*R(n,k+1)), n = 0,1,2,..., where R(n,x) = Sum_{k = 0..n} (-1)^k*binomial(n,k)*k!*2^(n-k)*binomial(-x,k) are the (unsigned) row polynomials of A137346. Cf. A094816. (End)

A098320 a(1)=0; for i>=1, a(i+1)=position of first occurrence of a(i) in decimal expansion of 1/e.

Original entry on oeis.org

0, 27, 88, 308, 267, 922, 811, 40, 150, 173, 555, 1751, 3389, 5859, 10579, 227865, 560966, 1382684, 12331649, 118447869

Offset: 0

Mark Hudson (mrmarkhudson(AT)hotmail.com), Sep 03 2004

Recurrence sequence based on positions of digits in decimal places of 1/e.

			So for example, a(2)=27 because 27th digit of 1/e after decimal point is 0.
a(3)=88 because 88th decimal digit of 1/e is where 27 appears,
a(4)=308 because 308th to 309th decimal digits of 1/e form "88" and so on.

Cf. A097614 for the analogous recurrence sequence for Pi, A098266 for e recurrence, A098289 for log(2) recurrence, A098290 for Zeta(3) recurrence, A098319 for 1/Pi recurrence. See A068985 for digits of 1/e.

More terms from Ben Ross (bmr180(AT)psu.edu), Feb 01 2006

A232745 Numbers k for which the largest m such that m! divides k is even.

Original entry on oeis.org

2, 4, 8, 10, 14, 16, 20, 22, 24, 26, 28, 32, 34, 38, 40, 44, 46, 48, 50, 52, 56, 58, 62, 64, 68, 70, 72, 74, 76, 80, 82, 86, 88, 92, 94, 96, 98, 100, 104, 106, 110, 112, 116, 118, 122, 124, 128, 130, 134, 136, 140, 142, 144, 146, 148, 152, 154, 158, 160, 164

Offset: 1

Antti Karttunen, Dec 01 2013

nonn

Numbers k for which A055881(k) is even.
Equally: Numbers k which have an odd number of the trailing zeros in their factorial base representation A007623(k).
The sequence can be described in the following manner: Sequence includes all multiples of 2! (even numbers), except that it excludes from those the multiples of 3! (6), except that it includes the multiples of 4! (24), except that it excludes the multiples of 5! (120), except that it includes the multiples of 6! (720), except that it excludes the multiples of 7! (5040), except that it includes the multiples of 8! (40320), except that it excludes the multiples of 9! (362880), except that it includes the multiples of 10! (3628800), except that ..., ad infinitum.
The number of terms not exceeding m! for m>=1 is A000166(m). The asymptotic density of this sequence is 1/e (A068985). - Amiram Eldar, Feb 26 2021

Antti Karttunen, Table of n, a(n) for n = 1..14833

Complement: A232744.
b(n) = A153880(A232744(n)) gives a subset of this sequence.
Cf. also A055881, A007623, A232741-A232743.
Analogous sequences for binary system: A003159 & A036554.
Cf. A000166, A068985.

seq[max_] := Select[Range[max!], OddQ @ LengthWhile[Reverse @ IntegerDigits[#, MixedRadix[Range[max, 2, -1]]], #1 == 0 &] &]; seq[5] (* Amiram Eldar, Feb 26 2021 *)

A196498 Decimal expansion of the constant Sum_{k>=0} (-1)^k/(10*k)!.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Oct 03 2011

			0.99999972442680776055212523675802047...

Michael I. Shamos, A catalog of the real numbers (2011).

Cf. A195070, A195394.

Cf. A068985, A049470, A346441, A346440, A346439, A346438, A346437, A346436, A346435.

RealDigits[ HypergeometricPFQ[{}, {1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10}, -10^-10], 10, 105] // First (* Jean-François Alcover, Feb 12 2013 *)

sumalt(k=0, (-1)^k/(10*k)!) \\ Michel Marcus, Jul 18 2021

6 more digits from Jean-François Alcover, Feb 12 2013

A346437 Decimal expansion of the constant Sum_{k>=0} (-1)^k/(7*k)!.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Jul 17 2021

			0.99980158731305804716545837...

Michael I. Shamos, A catalog of the real numbers (2011).

Cf. A068985, A049470, A346441, A346440, A346439, A346438, A346437, A346436, A346435, A196498.

RealDigits[HypergeometricPFQ[{}, {1/7, 2/7, 3/7, 4/7, 5/7, 6/7}, -1/7^7], 10, 120][[1]] (* Amiram Eldar, Jun 04 2023 *)

sumalt(k=0, (-1)^k/(7*k)!) \\ Michel Marcus, Jul 18 2021

A346440 Decimal expansion of the constant Sum_{k>=0} (-1)^k/(4*k)!.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Jul 17 2021

			0.95835813283300701621040446...

Michael I. Shamos, A catalog of the real numbers (2011).

Cf. A068985, A049470, A346441, A346439, A346438, A346437, A346436, A346435, A196498.

RealDigits[Sum[(-1)^k/(4*k)!, {k, 0, Infinity}], 10, 100][[1]] (* Amiram Eldar, Jul 18 2021 *)

sumalt(k=0, (-1)^k/(4*k)!) \\ Michel Marcus, Jul 18 2021

Equals cos(sqrt(2)/2)*cosh(sqrt(2)/2). - Amiram Eldar, Jul 18 2021

Continued fraction: 1/(1 + 1/(23 + 24/(1679 + ... + P(n-1)/((P(n) - 1) + ... )))), where P(n) = (4*n)*(4*n - 1)*(4*n - 2)*(4*n - 3) for n >= 1. Cf. A346441. - Peter Bala, Feb 21 2024

A061060 Write product of first n primes as x*y with x

Original entry on oeis.org

1, 1, 1, 1, 13, 17, 1, 41, 157, 1811, 1579, 18859, 95533, 17659, 1995293, 208303, 2396687, 58513111, 299808329, 2460653813, 3952306763, 341777053, 115405393057, 437621467859, 1009861675153, 6660853109087, 29075165225531

Offset: 1

Ed Pegg Jr, May 28 2001

nonn

			a(4)=1: 2*3*5*7 = 210 = 14*15, so we can take x=14, y=15, with difference of 1.
Also: n=3: 2*3-5=1; n=4: 3*5-2*7=1; n=5: 5*11-2*3*7=13; n=6: 2*7*13-3*5*11=17; n=7: 5*11*13-2*3*7*17=1; n=8: 3*5*11*19-2*7*13*17=41

Max Alekseyev, Table of n, a(n) for n = 1..70
C. Aebi and G. Cairns, Partitions of primes, Parabola 45, Issue 1 (2009). - _Jonathan Sondow_, Jun 21 2012
Carlos Rivera, Conjecture 18. Minimal Primorial Partitions, The Prime Puzzles & Problems Connection.

Cf. A038667, A060776, A060777, A061057, A060795, A060796, A061030-A061033, A182987, A263292, A352813.

A061060aux := proc(l1,l2) local resul ; resul := product(l1[i],i=1..nops(l1)) ; resul := resul-product(l2[i],i=1..nops(l2)) ; RETURN(abs(resul)) ; end:
A061060 := proc(n) local plist,i,subl,resul,j,l1,l2,k,d ; plist := [] ; resul := 1 ; for i from 1 to n do resul := resul*ithprime(i) ; plist := [op(plist), ithprime(i)] ; od; for i from 1 to n/2 do subl := combinat[choose](plist,i) ; for j from 1 to nops(subl) do l1 := op(j,subl) ; l2 := convert(plist,set) minus convert(l1,set) ; d := A061060aux(l1,l2) ; if d < resul then resul := d ; fi ; od; od ; RETURN(resul) ; end:
for n from 3 to 19 do printf("%d,",A061060(n)) ; od ; # R. J. Mathar, Aug 26 2006 [This Maple program was attached to A121315. However I think it belongs here, so I renamed the variables and moved it to this entry. - N. J. A. Sloane, Sep 16 2005]

(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) f[n_] := Block[{arrayofnprimes = Array[Prime, n], primorial = Times @@ Array[Prime, n], diffmin = Infinity, adiff, sub}, If[n == 1, 1, Do[sub = Times @@ NthSubset[i, arrayofnprimes]; adiff = Abs[primorial/sub - sub]; If[adiff < diffmin, diffmin = adiff], {i, 2, 2^n/2}]; diffmin]]; Do[ Print@f@n, {n, 30}] (* Robert G. Wilson v, Sep 14 2006 *)

Conjecture: Limit_{N->oo} (Sum_{n=1..N} log(a(n))) / (Sum_{n=1..N} prime(n)) = 1/e (A068985). - Alain Rocchelli, Nov 13 2023

Terms a(16)-a(45) in b-file computed by Jud McCranie, Apr 15 2000; Jan 12 2016
a(46)-a(60) in b-file from Don Reble, Jul 11 2020
a(61)-a(70) in b-file from Max Alekseyev, Apr 20 2022

A077613 Number of adjacent pairs of form (even,odd) among all permutations of {1,2,...,n}. Also, number of adjacent pairs of form (odd,even).

Original entry on oeis.org

0, 1, 4, 24, 144, 1080, 8640, 80640, 806400, 9072000, 108864000, 1437004800, 20118067200, 305124019200, 4881984307200, 83691159552000, 1506440871936000, 28810681675776000, 576213633515520000, 12164510040883200000, 267619220899430400000, 6182004002776842240000

Offset: 1

Leroy Quet, Frank Ruskey and Dean Hickerson, Nov 11 2002

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..449

Cf. A000142, A002620, A077611, A077612.

Cf. A001620, A068985, A099284.

Array[Floor[#/2] Ceiling[#/2] (# - 1)! &, 19] (* Michael De Vlieger, Aug 16 2017 *)

a(n) = floor(n/2)*ceil(n/2)*(n-1)!; \\ Michel Marcus, Aug 29 2013

a(n) = floor(n/2)*ceiling(n/2)*(n-1)!. Proof: There are floor(n/2)*ceiling(n/2) pairs (r, s) with r even and s odd. For each pair, there are n-1 places it can occur in a permutation and (n-2)! possible arrangements of the other numbers.
a(n) = A002620(n) * A000142(n-1). - Michel Marcus, Aug 29 2013
Sum_{n>=2} 1/a(n) = 6*(CoshIntegral(1) - gamma) + 2/e - 1 = 6*(A099284 - A001620) + 2*A068985 - 1. - Amiram Eldar, Jan 22 2023

A346435 Decimal expansion of the constant Sum_{k>=0} (-1)^k/(9*k)!.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Jul 17 2021

			0.99999724426807775760300443003850532493662547157742...

Michael I. Shamos, A catalog of the real numbers (2011).

Cf. A068985, A049470, A346441, A346440, A346439, A346438, A346437, A346436, A196498.

RealDigits[HypergeometricPFQ[{}, {1/9, 2/9, 1/3, 4/9, 5/9, 2/3, 7/9, 8/9}, -1/3^18], 10, 120][[1]] (* Amiram Eldar, Jun 04 2023 *)

sumalt(k=0, (-1)^k/(9*k)!) \\ Michel Marcus, Jul 18 2021

cp's OEIS Frontend

A072364 Decimal expansion of (1/e)^(1/e).

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A092553 Decimal expansion of 1/e^2.

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A098320 a(1)=0; for i>=1, a(i+1)=position of first occurrence of a(i) in decimal expansion of 1/e.

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A232745 Numbers k for which the largest m such that m! divides k is even.

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A196498 Decimal expansion of the constant Sum_{k>=0} (-1)^k/(10*k)!.

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A346437 Decimal expansion of the constant Sum_{k>=0} (-1)^k/(7*k)!.

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A346440 Decimal expansion of the constant Sum_{k>=0} (-1)^k/(4*k)!.

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A061060 Write product of first n primes as x*y with x

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