A290506 -id:A290506 - cp's OEIS frontend

A092605 Decimal expansion of e^(-1/2) or 1/sqrt(e).

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

Offset: 0

Mohammad K. Azarian, Apr 22 2004

For x = e^(-1/2), the largest prime factor of a random integer n is equally likely to be above or below n^x. - Charles R Greathouse IV, May 25 2009

Siegel's conjecture: this constant gives the density of regular primes among all the primes (see Ribenboim and Siegel). - Stefano Spezia, Apr 22 2025

			0.6065306597126334...

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 225.
C. L. Siegel, Zu zwei Bemerkungen Kummers. Nachr. Akad. d. Wiss. Göttingen, Math. Phys. Kl., II, 1964, 51-62. Reprinted in Gesammelte Abhandlungen (edited by K. Chandrasekharan and H. Maas), Vol. III, 436-442. Springer-Verlag, Berlin, 1966.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 502.
Index entries for transcendental numbers.

Cf. A000165, A001113, A019774, A068985, A290506.

RealDigits[E^-(1/2),10,120][[1]] (* Harvey P. Dale, Jul 23 2012 *)

exp(-.5) \\ Charles R Greathouse IV, Oct 02 2022

Equals Sum_{k>=0} (-1)^k/(2^k * k!) = Sum_{k>=0} (-1)^k/A000165(k). - Amiram Eldar, Aug 15 2020
From Peter Bala, Jan 16 2022; (Start)
Equals 16*Sum_{n >= 0} (-1)^n*n^2/((4*n^2 - 1)*(4*n^2 - 9)*(2^n)*n!).
Equals 8*Sum_{n >= 0} (-1)^n/(p(n)*p(n+1)*(2^n)*n!), where p(n) = 4*n^2 + 8*n + 1.
Equals 48*Sum_{n >= 0} (-1)^n/(q(n)*q(n+1)*(2^n)*n!), where q(n) = 8*n^3 + 36*n^2 + 34*n + 1. (End)
Equals i^(i/Pi), where i denotes the imaginary unit. - Stefano Spezia, Mar 01 2025
Equals 1 - A290506. - Amiram Eldar, Apr 22 2025

A120337 Euler-irregular primes p dividing E(2k) for some 2k < p-1.

Original entry on oeis.org

19, 31, 43, 47, 61, 67, 71, 79, 101, 137, 139, 149, 193, 223, 241, 251, 263, 277, 307, 311, 349, 353, 359, 373, 379, 419, 433, 461, 463, 491, 509, 541, 563, 571, 577, 587, 619, 677, 691, 709, 739, 751, 761, 769, 773, 811, 821, 877, 887, 907, 929, 941, 967, 971, 983

Offset: 1

Stefan Krämer, Jun 22 2006

nonn

Conjecture (Ernvall and Metsänkylä, 1978): The asymptotic density of this sequence within the primes is 1 - 1/sqrt(e) = 0.393469... (A290506), the same as the corresponding conjectured density of the irregular primes (A000928). - Amiram Eldar, Dec 06 2022

			a(1) = 19 because 19 divides E(10) = -19*2659 and 10 + 1 < 19.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
Reijo Ernvall, On the distribution mod 8 of the E-irregular primes, Annales Academiae Scientiarum Fennicae, Series A. I. Mathematica, Vol. 1, 1975, pp. 195-198.
Reijo Ernvall and Tauno Metsänkylä, Cyclotomic invariants and E-irregular primes, Mathematics of Computation, Vol. 32, No. 142 (1978), pp. 617-629; Corrigenda, ibid., Vol. 33, No. 145 (1979), p. 433.
Su Hu and Min-Soo Kim, A note on the irregular primes with respect to Euler polynomials, arXiv:1510.01558 [math.NT], 2015.
Su Hu, Min-Soo Kim, Pieter Moree and Min Sha, Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture, arXiv:1809.08431 [math.NT], 2018.
Romeo Mestrovic, A search for primes p such that Euler number E_{p-3} is divisible by p, arXiv preprint arXiv:1212.3602 [math.NT], 2012. - From _N. J. A. Sloane_, Jan 25 2013
Prime Pages, Euler Irregular
Samuel S. Wagstaff, Prime divisors of the Bernoulli and Euler numbers, Number theory for the millennium, III, 2002, pp. 357-374, 2002. MR 1956285.

Cf. A000928, A092218, A120115, A122045, A290506.

A120337_list := proc(bound)
local ae, F, p, m, maxp; F := NULL;
for m from 2 by 2 to bound do
  p := nextprime(m+1);
  ae := abs(euler(m));
  maxp := min(ae, bound);
  while p <= maxp do
      if ae mod p = 0
      then F := F,p fi;
      p := nextprime(p);
   od;
od;
sort([F]) end: # Peter Luschny, Apr 25 2011

fQ[p_] := Block[{k = 1}, While[ 2k +1 < p && Mod[ EulerE[ 2k], p] != 0, k++]; p > 2k +1]; Select[ Prime@ Range@ 168, fQ@# &] (* Robert G. Wilson v, Dec 10 2014 *)

The (trivial) divisors of E(2n) are given by the theorem of Sylvester (1861): Let p prime with p=1 (mod 4), p-1|2n, p^k|2n then p^{k+1} | E(2n).

Terms 251 through 983 from Peter Luschny, Apr 25 2011

A105018 Indices of the irregular primes.

Original entry on oeis.org

12, 17, 19, 26, 27, 32, 35, 37, 51, 55, 56, 58, 61, 62, 63, 64, 69, 71, 75, 77, 79, 80, 82, 84, 89, 90, 91, 94, 99, 100, 101, 102, 106, 107, 108, 111, 112, 113, 114, 115, 118, 119, 120, 122, 123, 124, 125, 129, 133, 134, 135, 137, 139, 140, 141, 142, 144, 146, 151

Offset: 1

Robert G. Wilson v, Mar 31 2005

nonn

Carl Ludwig Siegel, Zu zwei Bemerkungen Kummers, Nachrichten der Akademie der Wissenschaften in Göttingen, 1964, pp. 51-57.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Irregular Prime.

Cf. A000040, A000928, A105019, A290506.

fQ[n_] := Block[{ p = Prime[ n ], k = 1}, While[ 2k <= p - 3 && Mod[ Numerator[ BernoulliB[ 2k]], p] != 0, k++ ]; If[ 2k != p - 1, True, False]]; Select[ Range[2, 151], fQ[ # ] &]

Conjecture (Siegel, 1964): Limit_{n -> oo} a(n)/n = 1 - 1 / sqrt(e) (A290506). Of the 788060 primes < 12000000, 310443 are irregular.

A000040(a(n)) = A000928(n). - Amiram Eldar, Mar 06 2019

A306486 Number of squares in the interval [e^(n-1), e^n).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 13, 21, 36, 58, 96, 159, 262, 431, 712, 1172, 1934, 3189, 5256, 8667, 14289, 23559, 38841, 64039, 105583, 174076, 287003, 473188, 780155, 1286258, 2120681, 3496412, 5764609, 9504233, 15669832, 25835185, 42595018, 70227313, 115785266

Offset: 0

Alexei Kourbatov, Feb 18 2019

nonn

The lower endpoint e^(n-1) is included; the upper endpoint is not included. The terms a(0) to a(8) coincide with the Fibonacci numbers.

			Between exp(2) and exp(3) there are two squares, namely, 9 and 16; therefore, a(3)=2.

Alois P. Heinz, Table of n, a(n) for n = 0..4607 (first 501 terms from Alexei Kourbatov)

Cf. A000045, A000290, A001113, A005181, A019774, A290506, A306604.

a:= n-> (f-> f(n)-f(n-1))(i-> ceil(exp(i/2))):
seq(a(n), n=0..44);  # Alois P. Heinz, Feb 18 2019

a(n)=ceil(sqrt(exp(n)))-ceil(sqrt(exp(n-1)));
for(n=0,50,print1(a(n)", "))

a(n) = ceiling(sqrt(exp(n))) - ceiling(sqrt(exp(n-1))).
From Alois P. Heinz, Feb 19 2019: (Start)
Lim_{n->oo} a(n+1)/a(n) = sqrt(e) = 1.64872127... = A019774.
a(n) = A005181(n+1) - A005181(n). (End)
a(n) = (1-1/sqrt(e))*e^(n/2)+O(1) ~ 0.39346934...*e^(n/2) ~ A290506*e^(n/2). - Alexei Kourbatov, Feb 20 2019

cp's OEIS Frontend

A092605 Decimal expansion of e^(-1/2) or 1/sqrt(e).

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A120337 Euler-irregular primes p dividing E(2k) for some 2k < p-1.

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A105018 Indices of the irregular primes.

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A306486 Number of squares in the interval [e^(n-1), e^n).

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