A265237 -id:A265237 - cp's OEIS frontend

A122045 Euler (or secant) numbers E(n).

1, 0, -1, 0, 5, 0, -61, 0, 1385, 0, -50521, 0, 2702765, 0, -199360981, 0, 19391512145, 0, -2404879675441, 0, 370371188237525, 0, -69348874393137901, 0, 15514534163557086905, 0, -4087072509293123892361, 0, 1252259641403629865468285, 0, -441543893249023104553682821

Offset: 0

Roger L. Bagula, Sep 13 2006

sign

The convention in the OEIS is that the alternate zeros are normally omitted in such sequences. See A000364 for the official version of this sequence.

Odd primes p such that p | E(p-1) are primes p == 1 (mod 4), A002144. Conjecture: odd composites m such that m | E(m-1) are Carmichael numbers m such that p == 1 (mod 4) for every prime p|m, A265237. - Thomas Ordowski, Feb 06 2020

			G.f. = 1 - x^2 + 5*x^4 - 61*x^6 + 1385*x^8 - 50521*x^10 + 2702765*x^12 + ...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 5, page 41.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Fernando Barbero G., Juan Margalef-Bentabol, and Eduardo J. S. Villaseñor, A two-sided Faulhaber-like formula involving Bernoulli polynomials, arXiv:2002.00550 [math.NT], 2020.
Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo, and Matteo Silimbani, Ascending runs in permutations and valued Dyck paths, Ars Mathematica Contemporanea (2019) Vol. 16, No. 2, 445-463.
Hacène Belbachir and Yassine Otmani, Quadrinomial-Like Versions for Wolstenholme, Morley and Glaisher Congruences, Integers (2023) Vol. 23.
Fatima Zohra Bensaci, Rachid Boumahdi, and Laala Khaldi, Finite Sums Involving Fibonacci and Lucas Numbers, J. Int. Seq. (2024). See p. 2.
Michael D. Hirschhorn, Binomial Identities and Congruences for Euler Numbers, Fibonacci Quart. 53 (2015), no. 4, 319-322.
Guodong Liu, Generating functions and generalized Euler numbers, Proc. Japan Acad. Ser. A Math. Sci., Volume 84, Number 2 (2008), 29-34. See p 32.
F. Luca, A. Pizarro-Madariaga, and C. Pomerance, On the counting function of irregular primes, 2014.
Emanuele Munarini, Two-Parameter Identities for q-Appell Polynomials, Journal of Integer Sequences, Vol. 26 (2023), Article 23.3.1.
A. Niedermaier and J. Remmel, Analogues of up-down permutations for colored permutations, J. Int. Seq. 13 (2010) 10.5.6.
Bruce E. Sagan, Generalized Euler numbers and ordered set partitions, arXiv:2501.07692 [math.NT], 2025. See p. 2.
T. J. Stieltjes, Sur la réduction en fraction continue d'une série procédant suivant les puissances descendantes d'une variable, Ann. Fac. Sc. Toulouse 3 (1889) 1-17.

Cf. A000364, A002144, A028296, A265237, A331978, A350972.

m:=35; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( 1/Cosh(x) )); [Factorial(n-1)*b[n]: n in [1..m-1]]; // G. C. Greubel, Feb 13 2020

seq(euler(n) , n=0..31); # Zerinvary Lajos, Mar 15 2009
P := proc(n,x) option remember; if n = 0 then 1 else
   (n*x+(1/2)*(1-x))*P(n-1,x)+x*(1-x)*diff(P(n-1,x),x); expand(%) fi end:
A122045 := n -> (-1)^n*subs(x=-1, P(n,x)):
seq(A122045(n), n=0..30);  # Peter Luschny, Mar 07 2014
ptan := proc(n) option remember; if irem(n, 2) = 1 then 0 else
    -add(`if`(k=0, 1, binomial(n, k)*ptan(n - k)), k = 0..n-1,2) fi end:
A122045 := n -> ifelse(n = 0, 1, ptan(n)):
seq(A122045(n), n = 0..30); # Peter Luschny, Jun 06 2022

Table[EulerE[n], {n, 0, 30}]
Range[0, 30]! CoefficientList[ Series[ Sech[x], {x, 0, 30}], x] (* Robert G. Wilson v, Aug 08 2018 *)
a[0] := 1; a[n_?OddQ] := 0; a[n_?EvenQ] := a[n] = -Sum[a[k] Binomial[n, k], {k, 0, n - 1, 2}]; Map[a, Range[0, 30]] (* Oliver Seipel, May 20 2024 *)

a[n]:=if n<2 then 1-n else sum(-a[n-2*k]*binomial(n,2*k),k,1,floor(n/2));
makelist(a[n],n,0,50); /* Tani Akinari, Sep 15 2023 */

x='x+O('x^66); Vec(serlaplace(1/cosh(x))) \\ Joerg Arndt, Mar 10 2014

a(n) = 2^n*2^(n+1)*(subst(bernpol(n+1,x), x, 3/4) - subst(bernpol(n+1,x), x, 1/4))/(n+1); \\ Michel Marcus, May 20 2017

from sympy import bernoulli as B
def a(n): return int(2**n*2**(n + 1)*(B(n + 1, 3/4) - B(n + 1, 1/4))/(n + 1))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 24 2017, after PARI code by Michel Marcus

from functools import cache
from math import comb as binomial
@cache
def ptan(n):   # see also A331978 and A350972.
    return (0 if n % 2 == 1 else
    -sum(binomial(n,k) * ptan(n-k) if k > 0 else 1 for k in range(0, n-1, 2)))
def A122045(n): return 1 if n == 0 else ptan(n)
print([A122045(n) for n in range(31)])  # Peter Luschny, Jun 06 2022

[euler_number(i) for i in range(31)] # Zerinvary Lajos, Mar 15 2009

E.g.f.: sech(x). - Michael Somos, Mar 11 2014
a(n) = (Sum_{k>=0} (-1)^k*(2*k+1)^n)*2. - Gottfried Helms, Mar 09 2012
From Sergei N. Gladkovskii, Oct 14 2012 - Oct 13 2013: (Start)
Continued fractions:
G.f.: 1/U(0) where U(k) = 1 - x + x*(k+1)/(1 - x*(k+1)/U(k+1)).
G.f.: 1/U(0) where U(k) = 1 - x^2 + x^2*(2*k+1)*(2*k+2)/(1 + x^2*(2*k+1)*(2*k+2)/ U(k+1)).
E.g.f.: (1-x)/U(0) where U(k) = 1 - x/(1 - x/(x - (2*k+1)*(2*k+2)/U(k+1))).
E.g.f.: 1 - x^2/U(0) where U(k) = (2*k+1)*(2*k+2) + x^2 - x^2*(2*k+1)*(2*k+2)/U(k+1).
E.g.f.: 1/U(0) where U(k) = 1 + x^2/(2*(2*k+1)*(4*k+1) - 2*x^2*(2*k+1)*(4*k+1)/(x^2 + 4*(4*k+3)*(k+1)/U(k+1))).
E.g.f.: (2 + x^4/(U(0)*(x^2-2) - 2))/(2-x^2) where U(k) = 4*k + 4 + 1/(1 + x^2/(2 - x^2 + (2*k+3)*(2*k+4)/U(k+1))).
G.f.: 1/G(0) where G(k) = 1 + x^2*(2*k+1)^2/(1 + x^2*(2*k+2)^2/G(k+1)) (due to T. J. Stieltjes).
G.f.: 1/S(0) where S(k) = 1 + x^2*(k+1)^2/S(k+1) (due to T. J. Stieltjes).
G.f.: 1 - x/(1+x) + x/(1+x)/Q(0) where Q(k) = 1 - x + x*(k+2)/(1 - x*(k+1)/Q(k+1)).
G.f.: -(1/x)/Q(0) where Q(k) = -1/x + (k+1)^2/Q(k+1) (due to T. J. Stieltjes).
G.f.: (1/(1-x))/Q(0) + 1/(1-x) where Q(k) = 1 - 1/x + (k+1)*(k+2)/Q(k+1).
G.f.: (x/(x-1))/Q(0) + 1/(1-x) where Q(k) = 1 - x + x^2*(k+1)*(k+2)/Q(k+1).
G.f.: 1 - x/(1+x) + (x/(1+x))/Q(0) where Q(k) = 1 + x + (k+1)*(k+2)*x^2/Q(k+1).
E.g.f.: 1 - T(0)*x^2/(2+x^2) where T(k) = 1 - x^2*(2*k+1)*(2*k+2)/(x^2*(2*k+1)*(2*k+2) - ((2*k+1)*(2*k+2) + x^2)*((2*k+3)*(2*k+4) + x^2)/T(k+1)).
G.f.: T(0) where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2 + 1/T(k+1)). (End)
a(n) = 2^(2*n+1)*(zeta(-n,1/4) - zeta(-n,3/4)), where zeta(a, z) is the generalized Riemann zeta function. - Peter Luschny, Mar 11 2015
a(n) = 2^n*(2^(n+1)/(n+1))*(B(n+1, 3/4) - B(n+1, 1/4)) where B(n,x) is the n-th Bernoulli polynomial. See Liu link. - Michel Marcus, May 20 2017 [This is the same as: a(n) = -4^(n+1)*B(n+1, 1/4)*((n+1) mod 2)/(n+1). Peter Luschny, Oct 30 2020]
a(n) = 2*Im(PolyLog(-n, I)). - Peter Luschny, Sep 29 2020
a(4n) == 5 (mod 60) and a(4n+2) == -1 (mod 60). See Hirschhorn. - Michel Marcus, Jan 11 2022
For n > 1, a(n) = -Sum_{k=1..floor(n/2)} a(n-2*k)*binomial(n,2*k). - Tani Akinari, Sep 15 2023

Edited by N. J. A. Sloane, Sep 17 2006

A265285 Carmichael numbers (A002997) k such that k-1 is a square.

Original entry on oeis.org

46657, 2433601, 67371265, 351596817937, 422240040001, 18677955240001, 458631349862401, 286245437364810001, 20717489165917230086401

Offset: 1

Altug Alkan, Dec 06 2015

This sequence contains all Carmichael numbers n such that for all primes p dividing n, p-1 divides n-1 and furthermore, n-1 is a square.

Numbers sqrt(a(n)-1) form a subsequence of A135590. - Max Alekseyev, Apr 25 2024

			46657 is a term because 46657 - 1 = 46656 = 216^2.
2433601 is a term because 2433601 - 1 = 2433600 = 1560^2.

G. Tarry, I. Franel, A. Korselt, and G. Vacca, Problème chinois, L'intermédiaire des mathématiciens 6 (1899), pp. 142-144.
Eric Weisstein's World of Mathematics, Carmichael Number.
Index entries for sequences related to Carmichael numbers.

Subsequence of A265237 and of A265328.

Cf. A002997, A135590, A265237, A303791.

isA002997:= proc(n) local F,p;
         if n::even or isprime(n)  then return false fi;
         F:= ifactors(n)[2];
         if max(seq(f[2],f=F)) > 1 then return false fi;
         andmap(f -> (n-1) mod (f[1]-1) = 0,  F)
end proc:
select(isA002997, [seq(4*i^2+1,i=1..10^6)]); # Robert Israel, Dec 08 2015

is_c(n) = { my(f); bittest(n, 0) && !for(i=1, #f=factor(n)~, (f[2, i]==1 && n%(f[1, i]-1)==1)||return) && #f>1 }
for(n=1, 1e10, if(is_c(n) && issquare(n-1), print1(n, ", ")))

lista(kmax) = {my(m); for(k = 2, kmax, m = k^2 + 1; if(!isprime(m), f = factor(k); for(i = 1, #f~, f[i, 2] *= 2); fordiv(f, d, if(!(m % (d+1)) && isprime(d+1), m /= (d+1))); if(m == 1, print1(k^2 + 1, ", ")))); } \\ Amiram Eldar, May 02 2024

a(4)-a(5), using A002997 b-file, from Michel Marcus, Dec 07 2015
a(6) and a(7) from Robert Israel, Dec 08 2015
a(8) from Max Alekseyev, Apr 30 2018
a(9) from Daniel Suteu confirmed by Max Alekseyev, Apr 25 2024

A267462 Carmichael numbers that are not of the form x^2 + y^2 + z^2 where x, y and z are integers.

Original entry on oeis.org

8911, 1152271, 10267951, 14913991, 64377991, 67902031, 139952671, 178482151, 612816751, 652969351, 743404663, 2000436751, 2560600351, 3102234751, 3215031751, 5615659951, 5883081751, 7773873751, 8863329511, 9462932431, 10501586767, 11335174831, 12191597551, 13946829751, 16157879263, 21046047751

Offset: 1

Altug Alkan, Jan 15 2016

nonn

Intersection of A002997 and A004215.
Carmichael numbers that are the sum of 4 but no fewer nonzero squares.
Carmichael numbers of the form 8*k + 7.
Subsequence of A185321.
Carmichael numbers of the form x^2 + y^2 + z^2 where x, y and z are integers are 561, 1105, 1729, 2465, 2821, 6601, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973, 75361, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 278545, 294409, 314821, 334153, 340561, 399001, 410041, 449065, 488881, 512461, 530881, 552721, ...

			Carmichael number 561 is not a term of this sequence because 561 = 2^2 + 14^2 + 19^2.
Carmichael number 8911 is a term because there is no integer values of x, y and z for the equation 8911 = x^2 + y^2 + z^2.
Carmichael number 10585 is not a term because 10585 = 0^2 + 37^2 + 96^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
G. Tarry, I. Franel, A. Korselt, and G. Vacca. Problème chinois. L'intermédiaire des mathématiciens 6 (1899), pp. 142-144.
Eric Weisstein's World of Mathematics, Carmichael Number
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A004215, A185321, A265237.

filter:= proc(n)
  local q;
  if isprime(n) then return false fi;
  if 2 &^ (n-1) mod n <> 1 then return false fi;
  for q in ifactors(n)[2] do
    if q[2] > 1 or (n-1) mod (q[1]-1) <> 0 then return false fi
    od;
    true
end proc:
select(filter, [seq(8*k+7, k=0..10^7)]); # Robert Israel, Jan 18 2016

Select[8*Range[1,8000000]+7, CompositeQ[#] && Divisible[#-1, CarmichaelLambda[#]] &] (* Amiram Eldar, Jun 26 2019 *)

isA004215(n) = { my(fouri, j) ; fouri=1 ; while( n >=7*fouri, if( n % fouri ==0, j= n/fouri -7 ; if( j % 8 ==0, return(1) ) ; ) ; fouri *= 4 ; ) ; return(0) ; } { for(n=1, 400, if(isA004215(n), print1(n, ", ") ; ) ; ) ; }
isA002997(n) = { my(f); bittest(n, 0) && !for(i=1, #f=factor(n)~, (f[2, i]==1 && n%(f[1, i]-1)==1)||return) && #f>1 }
for(n=0, 1e10, if(isA002997(n) && isA004215(n), print1(n, ", ")));

isA002997(n) = { my(f); bittest(n, 0) && !for(i=1, #f=factor(n)~, (f[2, i]==1 && n%(f[1, i]-1)==1)||return) && #f>1 }
for(n=0, 1e10, if(isA002997(k=8*n+7), print1(k, ", ")));

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A122045 Euler (or secant) numbers E(n).

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A265285 Carmichael numbers (A002997) k such that k-1 is a square.

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Examples

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Crossrefs

Programs

Maple

PARI

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Extensions

A267462 Carmichael numbers that are not of the form x^2 + y^2 + z^2 where x, y and z are integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI