A027760 -id:A027760 - cp's OEIS frontend

A141056 1 followed by A027760, a variant of Bernoulli number denominators.

1, 2, 6, 2, 30, 2, 42, 2, 30, 2, 66, 2, 2730, 2, 6, 2, 510, 2, 798, 2, 330, 2, 138, 2, 2730, 2, 6, 2, 870, 2, 14322, 2, 510, 2, 6, 2, 1919190, 2, 6, 2, 13530, 2, 1806, 2, 690, 2, 282, 2, 46410, 2, 66, 2, 1590, 2, 798, 2, 870, 2, 354, 2, 56786730, 2, 6, 2, 510, 2, 64722, 2, 30, 2, 4686

Offset: 0

Paul Curtz, Aug 01 2008

nonn

The denominators of the Bernoulli numbers for n>0. B_n sequence begins 1, -1/2, 1/6, 0/2, -1/30, 0/2, 1/42, 0/2, ... This is an alternative version of A027642 suggested by the theorem of Clausen. - Peter Luschny, Apr 29 2009

Let f(n,k) = gcd { multinomial(n; n1, ..., nk) | n1 + ... + nk = n }; then a(n) = f(N,N-n+1)/f(N,N-n) for N >> n. - Mamuka Jibladze, Mar 07 2017

			The rational values as given by the e.g.f. in the formula section start: 1, 1/2, 7/6, 3/2, 59/30, 5/2, 127/42, 7/2, 119/30, ... - _Peter Luschny_, Aug 18 2018

Antti Karttunen, Table of n, a(n) for n = 0..10080
Thomas Clausen, Lehrsatz aus einer Abhandlung Über die Bernoullischen Zahlen, Astr. Nachr. 17 (22) (1840), 351-352.
Wikipedia, Bernoulli number

Cf. A027760, A027642, A176328.

Cf. A164555, A027642, A048993.

Clausen := proc(n) local S,i;
S := numtheory[divisors](n); S := map(i->i+1,S);
S := select(isprime,S); mul(i,i=S) end proc:
seq(Clausen(i),i=0..24);
# Peter Luschny, Apr 29 2009
A141056 := proc(n)
    if n = 0 then 1 else A027760(n) end if;
end proc: # R. J. Mathar, Oct 28 2013

a[n_] := Sum[ Boole[ PrimeQ[d+1]] / (d+1), {d, Divisors[n]}] // Denominator; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Aug 09 2012 *)

A141056(n) =
{
    p = 1;
    if (n > 0,
        fordiv(n, d,
            r = d + 1;
            if (isprime(r), p = p*r)
        )
    );
    return(p)
}
for(n=0,70,print1(A141056(n), ", ")); /* Peter Luschny, May 07 2012 */

a(n) are the denominators of the polynomials generated by cosh(x*z)*z/(1-exp(-z)) evaluated x=1. See A176328 for the numerators. - Peter Luschny, Aug 18 2018

a(n) = denominator(Sum_{j=0..n} (-1)^(n-j)*j!*Stirling2(n,j)*B(j)), where B are the Bernoulli numbers A164555/A027642. - Fabián Pereyra, Jan 06 2022

Extended by R. J. Mathar, Nov 22 2009

A317210 Composite numbers k+1 such that A002322(A027760(k)) = k.

Original entry on oeis.org

21, 45, 49, 81, 85, 91, 93, 111, 117, 121, 133, 141, 145, 165, 175, 177, 201, 205, 209, 213, 217, 221, 231, 235, 247, 253, 261, 265, 273, 289, 291, 301, 309, 319, 325, 333, 357, 361, 365, 369, 381, 391, 411, 415, 441, 445, 451, 453, 465, 469, 477, 481, 493

Offset: 1

Max Alekseyev and Thomas Ordowski, Jul 09 2018

nonn

Also, composite numbers n such that LCM( p-1 : prime p|A027642(n-1) ) = n-1. Also, composite numbers n such that LCM( p-1 : p is prime & (p-1)|(n-1) ) = n-1. - Max Alekseyev, Dec 03 2021

Contains all Carmichael numbers except 2628073, 3224065, 23382529, 182356993, 1419339691, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A027760, A002322.

1 + Select[Range[500], CompositeQ[# + 1] && CarmichaelLambda[ Times @@ Select[1 + Divisors@ #, PrimeQ]] == # &] (* Giovanni Resta, Aug 13 2018 *)

a027760(n) = denominator(sumdiv(n, d, if(isprime(d+1), 1/(d+1))));
a002322(n) = lcm(znstar(n)[2]);
isok(n) = !isprime(n) && (n--) && !frac(a002322(a027760(n))/n); \\ Michel Marcus, Aug 13 2018

More terms from Giovanni Resta, Aug 13 2018

A141410 Denominator of A027760(n+1)/n.

Original entry on oeis.org

1, 1, 2, 5, 3, 7, 4, 3, 5, 11, 6, 13, 7, 1, 8, 17, 9, 19, 10, 7, 11, 23, 12, 25, 13, 9, 14, 29, 15, 31, 16, 11, 17, 1, 18, 37, 19, 13, 20, 41, 21, 43, 22, 15, 23, 47, 24, 49, 25, 17, 26, 53, 27, 11, 28, 19, 29, 59, 30, 61

Offset: 2

Paul Curtz, Aug 04 2008

The series of fractions A027760(n+1)/n = 2/2, 30/3, 2/4, 42/5, 2/6, 30/7, 2/8, 66/9, 2/10, ... = 1, 10, 1/2, 42/5, 1/3, 30/7, 1/4, ... for n>=2 is represented here by the reduced denominators.

Cf. A141321.

A141410 := proc(n) A027760(n+1)/n ; denom(%) ; end proc:
seq(A141410(n),n=2..80) ; # R. J. Mathar, Jul 08 2011

a(2n) = n.

A141517 A141498(n)/A027760(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 47, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 1, 71, 73, 1, 1, 1, 5, 1, 7, 431, 1, 1, 1, 1, 97, 1, 1, 1, 1, 1, 1, 23, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 47, 1, 1, 1, 1, 1, 601, 1, 23, 1, 1, 5, 1, 1, 167, 7, 1, 431, 1, 1, 1, 1, 1, 1, 1, 1, 191, 193, 1

Offset: 1

Paul Curtz, Aug 11 2008

nonn

Composite entries are a(100)=25, a(120)=1205, a(144)=55969, a(156)=4069 etc.

A010696 := proc(n) op( (n mod 2)+1,[2,6]) ; end: A086892 := proc(n) gcd(2^n-1,3^n-1) ; end:
A141498 := proc(n) A010696(n-1)* A086892(n) ; end: A027760 := proc(n) local s, p; s := 0 ; p := 2; while p <= n+1 do if n mod (p-1) = 0 then s := s+1/p; fi; p := nextprime(p) ; od: denom(s) ; end:
A141517 := proc(n) A141498(n)/A027760(n) ; end: seq(A141517(n),n=1..120) ; # R. J. Mathar, Sep 03 2009

Offset modified, extended by R. J. Mathar, Sep 03 2009

A181904 a(n) = 2*(4^n - 1) / A027760(n).

Original entry on oeis.org

3, 5, 63, 17, 1023, 195, 16383, 4369, 262143, 31775, 4194303, 12291, 67108863, 89478485, 1073741823, 16843009, 17179869183, 172229265, 274877906943, 6663706835, 4398046511103, 254959218035, 70368744177663, 206208774147, 1125899906842623, 1501199875790165

Offset: 1

Peter Luschny, Apr 01 2012

nonn

The Clausen numbers A027760 divide 2*(4^n - 1).

Thomas Clausen, Lehrsatz aus einer Abhandlung ueber die Bernoulli'schen Zahlen, Astr. Nachr. 17 (1840), 351-352.
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoullischen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, Vol. 7 (1877), 157-187.

Cf. A002446, A027760

a[n_] := 2*(4^n-1)/Times @@ Select[Divisors[n]+1, PrimeQ]; Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Jul 29 2013 *)

def A181904(n) :
    def A027760(n) :
        S = divisors(n)
        S = map(lambda i: i+1, S)
        S = filter(lambda i: is_prime(i), S)
        return mul(i for i in S)
    return 2*(4^n-1) / A027760(n)
[A181904(n) for n in (1..26)]

A027642 Denominator of Bernoulli number B_n.

Original entry on oeis.org

1, 2, 6, 1, 30, 1, 42, 1, 30, 1, 66, 1, 2730, 1, 6, 1, 510, 1, 798, 1, 330, 1, 138, 1, 2730, 1, 6, 1, 870, 1, 14322, 1, 510, 1, 6, 1, 1919190, 1, 6, 1, 13530, 1, 1806, 1, 690, 1, 282, 1, 46410, 1, 66, 1, 1590, 1, 798, 1, 870, 1, 354, 1, 56786730, 1

Offset: 0

N. J. A. Sloane

Row products of A138243. - Mats Granvik, Mar 08 2008
From Gary W. Adamson, Aug 09 2008: (Start)
Equals row products of triangle A143343 and for a(n) > 1, row products of triangle A080092.
Julius Worpitzky's 1883 algorithm for generating Bernoulli numbers is described in A028246. (End)
The sequence of denominators of B_n is defined here by convention, not by necessity. The convention amounts to mapping 0 to the rational number 0/1. It might be more appropriate to regard numerators and denominators of the Bernoulli numbers as independent sequences N_n and D_n which combine to B_n = N_n / D_n. This is suggested by the theorem of Clausen which describes the denominators as the sequence D_n = 1, 2, 6, 2, 30, 2, 42, ... which combines with N_n = 1, -1, 1, 0, -1, 0, ... to the sequence of Bernoulli numbers. (Cf. A141056 and A027760.) - Peter Luschny, Apr 29 2009

			The sequence of Bernoulli numbers B_n (n = 0, 1, 2, ...) begins 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, 0, -3617/510, ... [Clarified by _N. J. A. Sloane_, Jun 02 2025]

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 810.
Jacob Bernoulli, Ars Conjectandi, Basel: Thurneysen Brothers, 1713. See page 97.
Thomas Clausen, "Lehrsatz aus einer Abhandlung Über die Bernoullischen Zahlen", Astr. Nachr. 17 (1840), 351-352 (see P. Luschny link).
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 106-108.
H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 230.
L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 137.
Roger Plymen, The Great Prime Number Race, AMS, 2020. See pp. 8-10.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 161.

T. D. Noe, Table of n, a(n) for n = 0..10000
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Beáta Bényi and Péter Hajnal, Poly-Bernoulli Numbers and Eulerian Numbers, arXiv:1804.01868 [math.CO], 2018.
Kwang-Wu Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.
Bakir Farhi, Formulas Involving Bernoulli and Stirling Numbers of Both Kinds, Journal of Integer Sequences, Vol. 28 (2025), Article 25.2.6. See p. 16.
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
H. W. Gould and Jocelyn Quaintance, Bernoulli Numbers and a New Binomial Transform Identity, J. Int. Seq. 17 (2014) # 14.2.2
Antal Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.
Masanobu Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
Guo-Dong Liu, H. M. Srivastava, and Hai-Quing Wang, Some Formulas for a Family of Numbers Analogous to the Higher-Order Bernoulli Numbers, J. Int. Seq. 17 (2014) # 14.4.6
Peter Luschny, Generalized Clausen numbers: definition and application.
Romeo Meštrović, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Antônio Francisco Neto, Carlitz's Identity for the Bernoulli Numbers and Zeon Algebra, J. Int. Seq. 18 (2015) # 15.5.6.
Ezgi Polat and Yilmaz Simsek, New formulas for Bernoulli polynomials with applications of matrix equations and Laplace transform, Pub. de l'Inst. Math. (2024) Vol. 116, Issue 130, 59-74. See p. 69.
Carl Pomerance and Samuel S. Wagstaff Jr, The denominators of the Bernoulli numbers, arXiv:2105.13252 [math.NT], 2021.
Jonathan Sondow and Emmanuel Tsukerman, The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers, arXiv:1401.0322 [math.NT], 2014; see section 5.
Matthew Roughan, The Polylogarithm Function in Julia, arXiv:2010.09860 [math.NA], 2020.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.
Wikipedia, Bernoulli number
Index entries for sequences related to Bernoulli numbers.
Index entries for "core" sequences

See A027641 (numerators) for full list of references, links, formulas, etc.
Cf. A002882, A003245, A127187, A127188, A138243, A028246, A143343, A080092, A141056, A027760.
Cf. A242179, A106831, A000142.
Cf. A001897, A002445, A277087.

a027642 n = a027642_list !! n
a027642_list = 1 : map (denominator . sum) (zipWith (zipWith (%))
   (zipWith (map . (*)) (tail a000142_list) a242179_tabf) a106831_tabf)
-- Reinhard Zumkeller, Jul 04 2014

[Denominator(Bernoulli(n)): n in [0..150]]; // Vincenzo Librandi, Mar 29 2011

(-1)^n*sum( (-1)^'m'*'m'!*stirling2(n,'m')/('m'+1),'m'=0..n);
A027642 := proc(n) denom(bernoulli(n)) ; end: # Zerinvary Lajos, Apr 08 2009

Table[ Denominator[ BernoulliB[n]], {n, 0, 68}] (* Robert G. Wilson v, Oct 11 2004 *)
Denominator[ Range[0, 68]! CoefficientList[ Series[x/(E^x - 1), {x, 0, 68}], x]]
(* Alternative code using Clausen Theorem: *)
A027642[k_Integer]:=If[EvenQ[k],Times@@Table[Max[1,Prime[i]*Boole[Divisible[k,Prime[i]-1]]],{i,1,PrimePi[2k]}],1+KroneckerDelta[k,1]]; (* Enrique Pérez Herrero, Jul 15 2010 *)
a[0] = 1; a[1] = 2; a[n_?OddQ] = 1; a[n_] := Times @@ Select[Divisors[n] + 1, PrimeQ]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 12 2012, after Ilan Vardi, when direct computation for large n is unfeasible *)

a(n)=if(n<0, 0, denominator(bernfrac(n)))

a(n) = if(n == 0 || (n > 1 && n % 2), 1, vecprod(select(x -> isprime(x), apply(x -> x + 1, divisors(n))))); \\ Amiram Eldar, Apr 24 2024

from sympy import bernoulli
[bernoulli(i).denominator for i in range(51)] # Indranil Ghosh, Mar 18 2017

def A027642_list(len):
    f, R, C = 1, [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        f *= n
        for k in range(n, 0, -1):
            C[k] = C[k-1] / (k+1)
        C[0] = -sum(C[k] for k in (1..n))
        R.append((C[0]*f).denominator())
    return R
A027642_list(62) # Peter Luschny, Feb 20 2016

E.g.f: x/(exp(x) - 1); take denominators.
Let E(x) be the e.g.f., then E(x) = U(0), where U(k) =  2*k + 1 - x*(2*k+1)/(x + (2*k+2)/(1 + x/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Jun 25 2012
E.g.f.: x/(exp(x)-1) = E(0) where E(k) = 2*k+1 - x/(2 + x/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 16 2013
E.g.f.: x/(exp(x)-1) = 2*E(0) - 2*x, where E(k)= x + (k+1)/(1 + 1/(1 - x/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 10 2013
E.g.f.: x/(exp(x)-1) = (1-x)/E(0), where E(k) = 1 - x*(k+1)/(x*(k+1) + (k+2-x)*(k+1-x)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 21 2013
E.g.f.: conjecture: x/(exp(x)-1) = T(0)/2 - x, where T(k) = 8*k+2 + x/( 1 - x/( 8*k+6 + x/( 1 - x/T(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2013
a(2*n) = 2*A001897(n) = A002445(n) = 3*A277087(n) for n >= 1. Jonathan Sondow, Dec 14 2016

A007850 Giuga numbers: composite numbers n such that p divides n/p - 1 for every prime divisor p of n.

Original entry on oeis.org

30, 858, 1722, 66198, 2214408306, 24423128562, 432749205173838, 14737133470010574, 550843391309130318, 244197000982499715087866346, 554079914617070801288578559178, 1910667181420507984555759916338506

Offset: 1

D. Borwein, J. M. Borwein, P. B. Borwein and R. Girgensohn

There are no other Giuga numbers with 8 or fewer prime factors. I did an exhaustive search using a PARI script which implemented Borwein and Girgensohn's method for finding n factor solutions given n - 2 factors. - Fred Schneider, Jul 04 2006
One further Giuga number is known with 10 prime factors, namely:
420001794970774706203871150967065663240419575375163060922876441614\
2557211582098432545190323474818 =
2 * 3 * 11 * 23 * 31 * 47059 * 2217342227 * 1729101023519 * 8491659218261819498490029296021 * 58254480569119734123541298976556403 but this may not be the next term. (See the Butske et al. paper.)
Conjecture: Giuga numbers are the solution of the differential equation n' = n + 1, where n' is the arithmetic derivative of n. - Paolo P. Lava, Nov 16 2009
n is a Giuga number if and only if n' = a*n + 1 for some integer a > 0 (see our preprint in arXiv:1103.2298). - José María Grau Ribas, Mar 19 2011
A composite number n is a Giuga number if and only if Sum_{i = 1..n-1} i^phi(n) == -1 (mod n), where phi(n) = A000010(n). - Jonathan Sondow, Jan 03 2014
A composite number n is a Giuga number if and only if Sum_{prime p|n} 1/p = 1/n + an integer. (In fact, all known Giuga numbers n satisfy Sum_{prime p|n} 1/p = 1/n + 1.) - Jonathan Sondow, Jan 08 2014
The prime factors of a(n) are listed as n-th row of A236434. - M. F. Hasler, Jul 13 2015
Conjecture: let k = a(n) and k be the product of x(n) distinct prime factors where x(n) <= x(n+1). Then, for any even n, n/2 + 2 <= x(n) <= n/2 + 3 and, for any odd n, (n+1)/2 + 2 <= x(n) <= (n+1)/2 + 3. For any n > 1, there are y "old" distinct prime factors o(1)...o(y) such that o(1) = 2, o(2) = 3, and z "new" distinct prime factors n(1)...n(z) such that none of them - unlike the "old" ones - can be a divisor of a(q) while q < n; n(1) > o(y), y = x(n) - z >= 2, 2 <= z <= b where b is either 4, or 1/2*n. - Sergey Pavlov, Feb 24 2017
Conjecture: a composite n is a Giuga number if and only if Sum_{k=1..n-1} k^lambda(n) == -1 (mod n), where lambda(n) = A002322(n). - Thomas Ordowski and Giovanni Resta, Jul 25 2018
A composite number n is a Giuga number if and only if A326690(n) = 1. - Jonathan Sondow, Jul 19 2019
A composite n is a Giuga number if and only if n * A027641(phi(n)) == - A027642(phi(n)) (mod n^2). Note: Euler's phi function A000010 can be replaced by the Carmichael lambda function A002322. - Thomas Ordowski, Jun 07 2020
By von Staudt and Clausen theorem, a composite n is a Giuga number if and only if n * A027759(phi(n)) == A027760(phi(n)) (mod n^2). Note: Euler's phi function can be replaced by the Carmichael lambda function. - Thomas Ordowski, Aug 01 2020

			From _M. F. Hasler_, Jul 13 2015: (Start)
The prime divisors of 30 are {2, 3, 5}, and 2 divides 30/2-1 = 14, 3 divides 30/3-1 = 9, and 5 divides 30/5-1 = 5.
The prime divisors of 858 are {2, 3, 11, 13} and 858/2-1 = 428 is even, 858/3-1 = 285 is divisible by 3, 858/11-1 = 77 is a multiple of 11, and 858/13-1 = 65 = 13*5.
(End)

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 30, pp 11, Ellipses, Paris 2008.

M. A. Alekseyev, J. M. Grau, A. M. Oller-Marcen. Computing solutions to the congruence 1^n + 2^n + ... + n^n == p (mod n). Discrete Applied Mathematics, 2018. doi:10.1016/j.dam.2018.05.022 arXiv:1602.02407 [math.NT], 2016.
D. Borwein, J. M. Borwein, P. B. Borwein and R. Girgensohn, Giuga's Conjecture on Primality, Amer. Math. Monthly 103, No. 1, 40-50 (1996).
J. M. Borwein and E. Wong, A Survey of Results Relating to Giuga's Conjecture on Primality, Vinet, Luc (ed.): Advances in Mathematical Sciences: CRM's 25 Years. Providence, RI: American Mathematical Society. CRM Proc. Lect. Notes. 11, 13-27 (1997).
William Butske, Lynda M. Jaje, and Daniel R. Mayernik, On the equation Sum_{p | N} 1/p + (1/N)=1, pseudoperfect numbers and perfectly weighted graphs, Math. Comp. 69 (2000), no. 229, 407-420.
José María Grau and Antonio M. Oller-Marcén, Giuga Numbers and the arithmetic derivative., arXiv:1103.2298 [math.NT], 2011; J. Int. Seq. 15 (2012) 12.4.1
José María Grau and Antonio M. Oller-Marcén, Generalizing Giuga's conjecture, arXiv:1103.3483 [math.NT], 2011.
J. M. Grau and A. M. Oller-Marcén, On the congruence sum_{j=1}^{n-1} j^{k(n-1)} == -1 (mod n); k-strong Giuga and k-Carmichael numbers, arXiv preprint arXiv:1311.3522 [math.NT], 2013.
J. M. Grau, A. M. Oller-Marcén, and D. Sadornil, On µ-Sondow Numbers, arXiv:2111.14211 [math.NT], 2021.
John Machacek, Egyptian Fractions and Prime Power Divisors, arXiv:1706.01008 [math.NT], 2017.
Mersenne Forum, Giuga numbers
Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867 [math.NT], May 04 2013.
R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
J. Sondow and K. MacMillan, Primary pseudoperfect numbers, arithmetic progressions, and the Erdős-Moser equation, Amer. Math. Monthly, 124 (2017) 232-240; arXiv:math/1812.06566 [math.NT], 2018.
J. Sondow and E. Tsukerman, The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers, arXiv:1401.0322 [math.NT], 2014; see section 4.
Eric Weisstein's World of Mathematics, Giuga Number.
Wikipedia, Agoh-Giuga conjecture
Wikipedia, Giuga number

Cf. A054377, A216823, A216824, A235137, A235138, A235140, A235363, A236434, A326690.

fQ[n_] := AllTrue[First /@ FactorInteger@ n, Divisible[n/# - 1, #] &]; Select[Range@ 100000, CompositeQ@ # && fQ@ # &] (* Michael De Vlieger, Oct 05 2015 *)

is(n)=if(isprime(n), return(0)); my(f=factor(n)[,1]); for(i=1,#f, if((n/f[i])%f[i]!=1, return(0))); n>1 \\ Charles R Greathouse IV, Apr 28 2015

from itertools import count, islice
from sympy import isprime, primefactors
def A007850_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda x: not isprime(x) and all((x//p-1) % p == 0 for p in primefactors(x)), count(max(startvalue,2)))
A007850_list = list(islice(A007850_gen(),4)) # Chai Wah Wu, Feb 19 2022

Sum_{i = 1..a(n)-1} i^phi(a(n)) == -1 (mod a(n)). - Jonathan Sondow, Jan 03 2014

a(12) from Fred Schneider, Jul 04 2006
Further references from Fred Schneider, Aug 19 2006
Definition corrected by Jonathan Sondow, Sep 16 2012

A091137 The Hirzebruch numbers. a(n) = Product_{2 <= p <= n+1, p prime} p^floor(n / (p - 1)).

Original entry on oeis.org

1, 2, 12, 24, 720, 1440, 60480, 120960, 3628800, 7257600, 479001600, 958003200, 2615348736000, 5230697472000, 31384184832000, 62768369664000, 32011868528640000, 64023737057280000, 51090942171709440000, 102181884343418880000, 33720021833328230400000, 67440043666656460800000

Offset: 0

Henry Bottomley, Dec 19 2003

nonn

Largest number m such that number of times m divides k! is almost k/n for large k, i.e., largest m with A090624(m) = n.
This is always a relatively small multiple of n!, since the multiplicity with which a prime p divides n! is always <= n/(p-1); it is equal to floor(n/(p-1)) at least when n is a power of p. - Franklin T. Adams-Watters, May 31 2010
At least for most small n, a(n) = A002790(n) * n!; the first difference is n=15. It appears that A002790(n) * n! always divides a(n).
Conjecture: The denominators of the series reversion of the sequence with e.g.f. Polylog(2,x). - Benedict W. J. Irwin, Jan 05 2017
Not only is a(n) divisible by n!; a(n) is divisible by (n + 1)! as has been observed by Bedhouche and Bakir (see links and A363596). - Hal M. Switkay, Aug 15 2025

			Let n = 4. The partitions of 4 are [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]. Thus a(4) = lcm([5, 4*2, 3*3, 3*2*2, 2*2*2*2]) = 720.

P. Curtz, Integration numérique ..., Note 12, C.C.S.A., Arcueil, 1969; see pp. 36, 56.
F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer, 3rd. ed., 1966; Lemma 1.7.3, p. 14. [From N. J. A. Sloane, Sep 06 2010]

Michael De Vlieger, Table of n, a(n) for n = 0..443
Abdelmalek Bedhouche and Bakir Farhi, On some products taken over the prime numbers, arXiv:2207.07957 [math.NT], 2022. See sigma_n p. 3.
Victor M. Buchstaber and Alexander P. Veselov, Todd polynomials and Hirzebruch numbers, arXiv:2310.07383 [math.AT], Oct. 2023.
Donghyun Kim and Jaeseong Oh, Extending the science fiction and the Loehr--Warrington formula, arXiv:2409.01041 [math.CO], 2024. See p. 32.

Starts similarly to A002207 especially for even n and all values of A002207 seen so far seem to divide a(n).

Cf. A002790, A000142, A090622, A090624, A091136, A171080, A238963, A275314, A368093, A368116, A363596.

A091137 := proc(n) local a,i,p ; a := 1 ; for i from 1 do p := ithprime(i) ; if p > n+1 then break; fi; a := a*p^floor(n/(p-1)) ; od: a ; end:
seq(A091137(n), n = 0..47); # R. J. Mathar, Feb 23 2009

A027760[n_] := Product[d, {d, Select[ Divisors[n] + 1, PrimeQ]}]; a[n_] := a[n] = A027760[n]*a[n-1]; a[0] = 1; Table[ a[n], {n, 0, 18}] (* Jean-François Alcover, Oct 04 2011 *)

a(n) = local(r); r=1; forprime(p=2, n+1, r*=p^(n\(p-1))); r
\\ Franklin T. Adams-Watters, May 31 2010

from math import prod
from sympy import primerange
def A091137(n): return prod(p**(n//(p-1)) for p in primerange(n+2))
# Chai Wah Wu, Apr 28 2023

def a(n): return lcm(product(r + 1 for r in p) for p in Partitions(n))
# Or, more efficient:
from functools import cache
@cache
def a_rec(n):
    if n == 0: return 1
    p = mul(s for s in map(lambda i: i + 1, divisors(n)) if is_prime(s))
    return p * a_rec(n - 1)
print([a_rec(n) for n in range(22)]) # Peter Luschny, Dec 12 2023

a(n) = Product_p {p prime} p^floor(n/(p-1)).
a(2n+1) = 2*a(2n).
a(n+1) = A027760(n+1)*a(n). - Paul Curtz, Aug 01 2008
From Peter Luschny, Dec 11 2023: (Start)
a(n) = lcm_{p in P(n)} Product_{r in p}(r + 1), where P(n) are the partitions of n.
a(n) = lcm(A238963row(n)).
a(n) = A368116(1, n), seen as the lcm of the product of the 1-shifted partitions.
a(n) = A368093(1, n), seen as the cumulative product of the Clausen numbers A160014(1, n).  (End)
a(n) = lcm({k: A275314(k) = n+1}). - Hal M. Switkay, Aug 13 2025
a(n) = (n + 1)! * A363596(n). - Hal M. Switkay, Aug 15 2025

New name using a formula of the author by Peter Luschny, Dec 11 2023

A160014 Generalized Clausen numbers (table read by antidiagonals).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 6, 3, 1, 2, 2, 3, 1, 1, 5, 30, 15, 5, 5, 1, 6, 2, 3, 1, 5, 1, 1, 7, 42, 21, 35, 35, 7, 7, 1, 2, 2, 15, 1, 5, 1, 7, 1, 1, 3, 30, 3, 5, 5, 7, 7, 1, 1, 1, 10, 2, 3, 1, 35, 1, 7, 1, 1, 1, 1, 11, 66, 165, 385, 55, 77, 77, 11, 11, 11, 11, 1

Offset: 0

Peter Luschny, Apr 29 2009

T(n,k) = Product_{ p - k | n} p, where p is prime.
T(n,0) is the squarefree kernel of n (A007947).
T(n,1) are the classical Clausen numbers (A141056). The classical Clausen numbers are by the von Staudt-Clausen theorem the denominators of the Bernoulli numbers.

			[k\n][0--1--2---3---4---5---6---7----8----9---10---11----12---13---14----15]
[0]...1..1..2...3...2...5...6...7....2....3...10...11.....6...13...14....15
[1]...1..2..6...2..30...2..42...2...30....2...66....2..2730....2....6.....2
[2]...1..3..3..15...3..21..15...3....3..165...21...39....15....3....3..1785
[3]...1..1..5...1..35...1...5...1..385....1...65....1....35....1...85.....1
[4]...1..5..5..35...5...5..35..55....5..455....5....5....35...85...55...665
[5]...1..1..7...1...7...1..77...1...91....1....7....1..1309....1..133.....1
T(3,4) = 35 = 5*7 because 5 and 7 are the only prime numbers p such that
(p - 4) divides 3.

Clausen, Thomas, "Lehrsatz aus einer Abhandlung ueber die Bernoullischen Zahlen", Astr. Nachr. 17 (1840), 351-352.

Charles R Greathouse IV, Rows n = 0..100, flattened
A. Hurwitz, Über die Entwicklungskoeffizienten der lemniskatischen Funktionen, Math. Ann., 51 (1899), 196-226; Mathematische Werke. Vols. 1 and 2, Birkhäuser, Basel, 1962-1963, see Vol. 2, No. LXVII.
Peter Luschny, Generalized Bernoulli numbers.

Cf. A007947, A141056, A027760, A027642.

Clausen := proc(n,k) local S,i;
S := numtheory[divisors](n);
S := map(i->i+k,S);
S := select(isprime,S);
mul(i,i=S) end:

t[0, ] = 1; t[n, k_] := Times @@ (Select[Divisors[n], PrimeQ[# + k] &] + k); Table[t[n-k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 26 2013 *)

T(n,k)=if(n,my(s=1);fordiv(n,d,if(isprime(d+k),s*=d+k)); s, 1)
for(s=0,9,for(k=0,s,print1(T(s-k,k)", "))) \\ Charles R Greathouse IV, Jun 26 2013

def Clausen(n, k):
    if k == 0: return 1
    return mul(filter(lambda s: is_prime(s), map(lambda i: i+n, divisors(k))))
for n in (0..5): [Clausen(n, k) for k in (0..15)]   # Peter Luschny, Jun 05 2013

Swapped n<>k fixed by Peter Luschny, May 04 2009

A249134 Numbers k such that Bernoulli number B_k has denominator 2730.

Original entry on oeis.org

12, 24, 1308, 1884, 2004, 2364, 2532, 2724, 3804, 4008, 4044, 4188, 4236, 4668, 5052, 5064, 5268, 5388, 5484, 6252, 6492, 6564, 6756, 6852, 7044, 7188, 7356, 7404, 7608, 7764, 8124, 8412, 8472, 8796, 9084, 9228, 9852, 9876, 9924

Offset: 1

Jean-François Alcover and Paul Curtz, Oct 22 2014

nonn

2730 = 2 * 3 * 5 * 7 * 13.

			BernoulliB(12) is -691/2730, hence 12 is in the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000367, A002445, A006954, A027642, A027760, A027762, A051222, A051225-A051230, A245056, A271634, A271635, A272138, A272139, A272140, A272183, A272184, A272185, A272186.

Reap[For[n = 0, n <= 10^4, n = n+12, If[Denominator[BernoulliB[n]] == 2730, Print[n]; Sow[n]]]][[2, 1]]
Select[Table[n, {n, 2, 10000}], Denominator@BernoulliB[#]==2730 &] (* Vincenzo Librandi, Apr 02 2015 *)

is(n)=denominator(bernfrac(n))==2730 \\ Charles R Greathouse IV, Oct 22 2014

is(n)=if(n%12 || n%16==0 || n%9==0, return(0)); forprime(p=5,107, if(n%p==0, return(0))); fordiv(n,d, if(isprime(d+1) && d>13, return(0))); 1 \\ Charles R Greathouse IV, Oct 22 2014

cp's OEIS Frontend

A141056 1 followed by A027760, a variant of Bernoulli number denominators.

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A317210 Composite numbers k+1 such that A002322(A027760(k)) = k.

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A141410 Denominator of A027760(n+1)/n.

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A141517 A141498(n)/A027760(n).

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A181904 a(n) = 2*(4^n - 1) / A027760(n).

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A027642 Denominator of Bernoulli number B_n.

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A007850 Giuga numbers: composite numbers n such that p divides n/p - 1 for every prime divisor p of n.

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