A083207 -id:A083207 - cp's OEIS frontend

A001710 Order of alternating group A_n, or number of even permutations of n letters.

1, 1, 1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400, 239500800, 3113510400, 43589145600, 653837184000, 10461394944000, 177843714048000, 3201186852864000, 60822550204416000, 1216451004088320000, 25545471085854720000, 562000363888803840000

Offset: 0

N. J. A. Sloane

For n >= 3, a(n-1) is also the number of ways that a 3-cycle in the symmetric group S_n can be written as a product of 2 long cycles (of length n). - Ahmed Fares (ahmedfares(AT)my-deja.com), Aug 14 2001
a(n) is the number of Hamiltonian circuit masks for an n X n adjacency matrix of an undirected graph. - Chad Brewbaker, Jan 31 2003
a(n-1) is the number of necklaces one can make with n distinct beads: n! bead permutations, divide by two to represent flipping the necklace over, divide by n to represent rotating the necklace. Related to Stirling numbers of the first kind, Stirling cycles. - Chad Brewbaker, Jan 31 2003
Number of increasing runs in all permutations of [n-1] (n>=2). Example: a(4)=12 because we have 12 increasing runs in all the permutations of [3] (shown in parentheses): (123), (13)(2), (3)(12), (2)(13), (23)(1), (3)(2)(1). - Emeric Deutsch, Aug 28 2004
Minimum permanent over all n X n (0,1)-matrices with exactly n/2 zeros. - Simone Severini, Oct 15 2004
The number of permutations of 1..n that have 2 following 1 for n >= 1 is 0, 1, 3, 12, 60, 360, 2520, 20160, ... . - Jon Perry, Sep 20 2008
Starting (1, 3, 12, 60, ...) = binomial transform of A000153: (1, 2, 7, 32, 181, ...). - Gary W. Adamson, Dec 25 2008
First column of A092582. - Mats Granvik, Feb 08 2009
The asymptotic expansion of the higher order exponential integral E(x,m=1,n=3) ~ exp(-x)/x*(1 - 3/x + 12/x^2 - 60/x^3 + 360/x^4 - 2520/x^5 + 20160/x^6 - 81440/x^7 + ...) leads to the sequence given above. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009
For n>1: a(n) = A173333(n,2). - Reinhard Zumkeller, Feb 19 2010
Starting (1, 3, 12, 60, ...) = eigensequence of triangle A002260, (a triangle with k terms of (1,2,3,...) in each row given k=1,2,3,...). Example: a(6) = 360, generated from (1, 2, 3, 4, 5) dot (1, 1, 3, 12, 60) = (1 + 2 + 9 + 48 + 300). - Gary W. Adamson, Aug 02 2010
For n>=2: a(n) is the number of connected 2-regular labeled graphs on (n+1) nodes (Cf. A001205). - Geoffrey Critzer, Feb 16 2011.
The Fi1 and Fi2 triangle sums of A094638 are given by the terms of this sequence (n>=1). For the definition of these triangle sums see A180662. - Johannes W. Meijer, Apr 20 2011
Also [1, 1] together with the row sums of triangle A162608. - Omar E. Pol, Mar 09 2012
a(n-1) is, for n>=2, also the number of necklaces with n beads (only C_n symmetry, no turnover) with n-1 distinct colors and signature c[.]^2 c[.]^(n-2). This means that two beads have the same color, and for n=2 the second factor is omitted. Say, cyclic(c[1]c[1]c[2]c[3]..c[n-1]), in short 1123...(n-1), taken cyclically. E.g., n=2: 11,  n=3: 112, n=4: 1123, 1132, 1213,  n=5: 11234, 11243, 11324, 11342, 11423, 11432, 12134, 12143, 13124, 13142, 14123, 14132. See the next-to-last entry in line n>=2 of the representative necklace partition array A212359. - Wolfdieter Lang, Jun 26 2012
For m >= 3, a(m-1) is the number of distinct Hamiltonian circuits in a complete simple graph with m vertices. See also A001286. - Stanislav Sykora, May 10 2014
In factorial base (A007623) these numbers have a simple pattern: 1, 1, 1, 11, 200, 2200, 30000, 330000, 4000000, 44000000, 500000000, 5500000000, 60000000000, 660000000000, 7000000000000, 77000000000000, 800000000000000, 8800000000000000, 90000000000000000, 990000000000000000, etc. See also the formula based on this observation, given below. - Antti Karttunen, Dec 19 2015
Also (by definition) the independence number of the n-transposition graph. - Eric W. Weisstein, May 21 2017
Number of permutations of n letters containing an even number of even cycles. - Michael Somos, Jul 11 2018
Equivalent to Brewbaker's and Sykora's comments, a(n - 1) is the number of undirected cycles covering n labeled vertices, hence the logarithmic transform of A002135. - Gus Wiseman, Oct 20 2018
For n >= 2 and a set of n distinct leaf labels, a(n) is the number of binary, rooted, leaf-labeled tree topologies that have a caterpillar shape (column k=1 of A306364). - Noah A Rosenberg, Feb 11 2019
Also the clique covering number of the n-Bruhat graph. - Eric W. Weisstein, Apr 19 2019
a(n) is the number of lattices of the form [s,w] in the weak order on S_n, for a fixed simple reflection s. - Bridget Tenner, Jan 16 2020
For n > 3, a(n) = p_1^e_1*...*p_m^e_m, where p_1 = 2 and e_m = 1. There exists p_1^x where x <= e_1 such that p_1^x*p_m^e_m is a primitive Zumkeller number (A180332) and p_1^e_1*p_m^e_m is a Zumkeller number (A083207). Therefore, for n > 3, a(n) = p_1^e_1*p_m^e_m*r, where r is relatively prime to p_1*p_m, is also a Zumkeller number. - Ivan N. Ianakiev, Mar 11 2020
For n>1, a(n) is the number of permutations of [n] that have 1 and 2 as cycle-mates, that is, 1 and 2 are contained in the same cycle of a cyclic representation of permutations of [n]. For example, a(4) counts the 12 permutations with 1 and 2 as cycle-mates, namely, (1 2 3 4), (1 2 4 3), (1 3 2 4), (1 3 4 2), (1 4 2 3), (1 4 3 2), (1 2 3) (4), (1 3 2) (4), (1 2 4 )(3), (1 4 2)(3), (1 2)(3 4), and (1 2)(3)(4). Since a(n+2)=row sums of A162608, our result readily follows. - Dennis P. Walsh, May 28 2020

			G.f. = 1 + x + x^2 + 3*x^3 + 12*x^4 + 60*x^5 + 360*x^6 + 2520*x^7 + ...

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 87-8, 20. (a), c_n^e(t=1).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..100
Somaya Barati, Beáta Bényi, Abbas Jafarzadeh, and Daniel Yaqubi, Mixed restricted Stirling numbers, arXiv:1812.02955 [math.CO], 2018.
Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.
Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
Jonathan Beagley and Lara Pudwell, Colorful Tilings and Permutations, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.4.
Olivier Bodini, Antoine Genitrini, and Mehdi Naima, Ranked Schröder Trees, arXiv:1808.08376 [cs.DS], 2018.
Olivier Bodini, Antoine Genitrini, Cécile Mailler, and Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), Article 00.1.5.
Camille Combe and Samuele Giraudo, Cliff operads: a hierarchy of operads on words, arXiv:2106.14552 [math.CO], 2021.
Mareike Fischer, Extremal Values of the Sackin Tree Balance Index, Ann. Comb. (2021) Vol. 25, 515-541, Remark 7.
Hannah Golab, Pattern avoidance in Cayley permutations, Master's Thesis, Northern Arizona Univ. (2024). See p. 36.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 262.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
Shirali Kadyrov and Farukh Mashurov, Generalized continued fraction expansions for Pi and E, arXiv:1912.03214 [math.NT], 2019.
Chanchal Kumar and Amit Roy, Integer Sequences and Monomial Ideals, arXiv:2003.10098 [math.CO], 2020.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
Xah Lee, Combinatorics: Loop in n points.
D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
S-Z Song, S-G Hwang, S-H Rim, and G-S Cheon, Extremes of permanents of (0,1)-matrices, Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002). Linear Algebra Appl. 373 (2003), 197-210.
A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
B. E. Tenner, Interval structures in the Bruhat and weak orders, arXiv:2001.05011 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Alternating Group.
Eric Weisstein's World of Mathematics, Bruhat Graph.
Eric Weisstein's World of Mathematics, Circular Permutation.
Eric Weisstein's World of Mathematics, Clique Covering Number.
Eric Weisstein's World of Mathematics, Even Permutation.
Eric Weisstein's World of Mathematics, Hamiltonian Cycle.
Eric Weisstein's World of Mathematics, Independence Number.
Eric Weisstein's World of Mathematics, Odd Permutation.
Eric Weisstein's World of Mathematics, Transposition Graph.
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 7.
Index entries for sequences related to factorial base representation.
Index entries for sequences related to factorial numbers.
Index entries for sequences related to groups.
Index to divisibility sequences.

Cf. A000142, A000153, A000255, A001147, A001286, A001720, A002135, A002260, A007623, A007717, A049444, A049459, A093468, A094587, A094638, A138533, A153880, A173333, A213936, A215771, A319225, A319226, A320655.
a(n+1)= A046089(n, 1), n >= 1 (first column of triangle), A161739 (q(n) sequence).
Bisections are A002674 and A085990 (essentially).
Row 3 of A265609 (essentially).
Row sums of A307429.

[1] cat [Order(AlternatingGroup(n)): n in [1..20]]; // Arkadiusz Wesolowski, May 17 2014

seq(mul(k, k=3..n), n=0..20); # Zerinvary Lajos, Sep 14 2007

a[n_]:= If[n > 2, n!/2, 1]; Array[a, 21, 0]
a[n_]:= If[n<3, 1, n*a[n-1]]; Array[a, 21, 0]; (* Robert G. Wilson v, Apr 16 2011 *)
a[ n_]:= If[n<0, 0, n! SeriesCoefficient[(2-x^2)/(2-2x), {x, 0, n}]]; (* Michael Somos, May 22 2014 *)
a[ n_]:= If[n<0, 0, n! SeriesCoefficient[1 +Sinh[-Log[1-x]], {x, 0, n}]]; (* Michael Somos, May 22 2014 *)
Numerator[Range[0, 20]!/2] (* Eric W. Weisstein, May 21 2017 *)
Table[GroupOrder[AlternatingGroup[n]], {n, 0, 20}] (* Eric W. Weisstein, May 21 2017 *)

PARI
```
{a(n) = if( n<2, n>=0, n!/2)};
```

a(n)=polcoeff(1+x*sum(m=0,n,m^m*x^m/(1+m*x+x*O(x^n))^m),n) \\ Paul D. Hanna

A001710=n->n!\2+(n<2) \\ M. F. Hasler, Dec 01 2013

from math import factorial
def A001710(n): return factorial(n)>>1 if n > 1 else 1 # Chai Wah Wu, Feb 14 2023

def A001710(n): return (factorial(n) +int(n<2))//2
[A001710(n) for n in range(31)] # G. C. Greubel, Sep 28 2024

;; Using memoization-macro definec for which an implementation can be found in http://oeis.org/wiki/Memoization
(definec (A001710 n) (cond ((<= n 2) 1) (else (* n (A001710 (- n 1))))))
;; Antti Karttunen, Dec 19 2015

a(n) = numerator(n!/2) and A141044(n) = denominator(n!/2).
D-finite with recurrence: a(0) = a(1) = a(2) = 1; a(n) = n*a(n-1) for n>2. - Chad Brewbaker, Jan 31 2003 [Corrected by N. J. A. Sloane, Jul 25 2008]
a(0) = 0, a(1) = 1; a(n) = Sum_{k=1..n-1} k*a(k). - Amarnath Murthy, Oct 29 2002
Stirling transform of a(n+1) = [1, 3, 12, 160, ...] is A083410(n) = [1, 4, 22, 154, ...]. - Michael Somos, Mar 04 2004
First Eulerian transform of A000027. See A000142 for definition of FET. - Ross La Haye, Feb 14 2005
From Paul Barry, Apr 18 2005: (Start)
a(n) = 0^n + Sum_{k=0..n} (-1)^(n-k-1)*T(n-1, k)*cos(Pi*(n-k-1)/2)^2.
T(n,k) = abs(A008276(n, k)). (End)
E.g.f.: (2 - x^2)/(2 - 2*x).
E.g.f. of a(n+2), n>=0, is 1/(1-x)^3.
E.g.f.: 1 + sinh(log(1/(1-x))). - Geoffrey Critzer, Dec 12 2010
a(n+1) = (-1)^n * A136656(n,1), n>=1.
a(n) = n!/2 for n>=2 (proof from the e.g.f). - Wolfdieter Lang, Apr 30 2010
a(n) = (n-2)! * t(n-1), n>1, where t(n) is the n-th triangular number (A000217). - Gary Detlefs, May 21 2010
a(n) = ( A000254(n) - 2* A001711(n-3) )/3, n>2. - Gary Detlefs, May 24 2010
O.g.f.: 1 + x*Sum_{n>=0} n^n*x^n/(1 + n*x)^n. - Paul D. Hanna, Sep 13 2011
a(n) = if n < 2 then 1, otherwise Pochhammer(n,n)/binomial(2*n,n). - Peter Luschny, Nov 07 2011
a(n) = Sum_{k=0..floor(n/2)} s(n,n-2*k) where s(n,k) are Stirling number of the first kind, A048994. - Mircea Merca, Apr 07 2012
a(n-1), n>=3, is M_1([2,1^(n-2)])/n = (n-1)!/2, with the M_1 multinomial numbers for the given n-1 part partition of n. See the second to last entry in line n>=3 of A036038, and the above necklace comment by W. Lang. - Wolfdieter Lang, Jun 26 2012
G.f.: A(x) = 1 + x + x^2/(G(0)-2*x) where G(k) = 1 - (k+1)*x/(1 - x*(k+3)/G(k+1)); (continued fraction). - Sergei N. Gladkovskii, Dec 26 2012.
G.f.: 1 + x + (Q(0)-1)*x^2/(2*(sqrt(x)+x)), where Q(k) = 1 + (k+2)*sqrt(x)/(1 - sqrt(x)/(sqrt(x) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
G.f.: 1 + x + (x*Q(x)-x^2)/(2*(sqrt(x)+x)), where Q(x) = Sum_{n>=0} (n+1)!*x^n*sqrt(x)*(sqrt(x) + x*(n+2)). - Sergei N. Gladkovskii, May 15 2013
G.f.: 1 + x/2 + (Q(0)-1)*x/(2*(sqrt(x)+x)), where Q(k) = 1 + (k+1)*sqrt(x)/(1 - sqrt(x)/(sqrt(x) + 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013
G.f.: 1 + x + x^2*G(0)/2, where G(k) = 1 + 1/(1 - x/(x + 1/(k+3)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
G.f.: 1+x + x^2*W(0), where W(k) = 1 - x*(k+3)/( x*(k+3) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013
From Antti Karttunen, Dec 19 2015: (Start)
a(0)=a(1)=1; after which, for even n: a(n) = (n/2) * (n-1)!, and for odd n: a(n) = (n-1)/2 * ((n-1)! + (n-2)!). [The formula was empirically found after viewing these numbers in factorial base, A007623, and is easily proved by considering formulas from Lang (Apr 30 2010) and Detlefs (May 21 2010) shown above.]
For n >= 1, a(2*n+1) = a(2*n) + A153880(a(2*n)). [Follows from above.] (End)
Inverse Stirling transform of a(n) is (-1)^(n-1)*A009566(n). - Anton Zakharov, Aug 07 2016
a(n) ~ sqrt(Pi/2)*n^(n+1/2)/exp(n). - Ilya Gutkovskiy, Aug 07 2016
a(n) = A006595(n-1)*n/A000124(n) for n>=2. - Anton Zakharov, Aug 23 2016
a(n) = A001563(n-1) - A001286(n-1) for n>=2. - Anton Zakharov, Sep 23 2016
From Peter Bala, May 24 2017: (Start)
The o.g.f. A(x) satisfies the Riccati equation x^2*A'(x) + (x - 1)*A(x) + 1 - x^2 = 0.
G.f.: A(x) = 1 + x + x^2/(1 - 3*x/(1 - x/(1 - 4*x/(1 - 2*x/(1 - 5*x/(1 - 3*x/(1  - ... - (n + 2)*x/(1 - n*x/(1 - ... ))))))))) (apply Stokes, 1982).
A(x) = 1 + x + x^2/(1 - 2*x - x/(1 - 3*x/(1 - 2*x/(1 - 4*x/(1 - 3*x/(1 - 5*x/(1 - ... - n*x/(1 - (n+2)*x/(1 - ... ))))))))). (End)
H(x) = (1 - (1 + x)^(-2)) / 2 = x - 3*x^2/2! + 12*x^3/3! - ..., an e.g.f. for the signed sequence here (n!/2!), ignoring the first two terms, is the compositional inverse of G(x) = (1 - 2*x)^(-1/2) - 1 = x + 3*x^2/2! + 15*x^3/3! + ..., an e.g.f. for A001147. Cf. A094638. H(x) is the e.g.f. for the sequence (-1)^m * m!/2 for m = 2,3,4,... . Cf. A001715 for n!/3! and A001720 for n!/4!. Cf. columns of A094587, A173333, and A213936 and rows of A138533. - Tom Copeland, Dec 27 2019
From Amiram Eldar, Jan 08 2023: (Start)
Sum_{n>=0} 1/a(n) = 2*(e-1).
Sum_{n>=0} (-1)^n/a(n) = 2/e. (End)

More terms from Larry Reeves (larryr(AT)acm.org), Aug 20 2001

Further terms from Simone Severini, Oct 15 2004

A001813 Quadruple factorial numbers: a(n) = (2n)!/n!.

Original entry on oeis.org

1, 2, 12, 120, 1680, 30240, 665280, 17297280, 518918400, 17643225600, 670442572800, 28158588057600, 1295295050649600, 64764752532480000, 3497296636753920000, 202843204931727360000, 12576278705767096320000, 830034394580628357120000, 58102407620643984998400000

Offset: 0

N. J. A. Sloane

Counts binary rooted trees (with out-degree <= 2), embedded in plane, with n labeled end nodes of degree 1. Unlabeled version gives Catalan numbers A000108.
Define a "downgrade" to be the permutation which places the items of a permutation in descending order. We are concerned with permutations that are identical to their downgrades. Only permutations of order 4n and 4n+1 can have this property; the number of permutations of length 4n having this property are equinumerous with those of length 4n+1. If a permutation p has this property then the reversal of this permutation also has it. a(n) = number of permutations of length 4n and 4n+1 that are identical to their downgrades. - Eugene McDonnell (eemcd(AT)mac.com), Oct 26 2003
Number of broadcast schemes in the complete graph on n+1 vertices, K_{n+1}. - Calin D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008
Hankel transform is A137565. - Paul Barry, Nov 25 2009
The e.g.f. of 1/a(n) = n!/(2*n)! is (exp(sqrt(x)) + exp(-sqrt(x)) )/2. - Wolfdieter Lang, Jan 09 2012
From Tom Copeland, Nov 15 2014: (Start)
Aerated with intervening zeros (1,0,2,0,12,0,120,...) = a(n) (cf. A123023 and A001147), the e.g.f. is e^(t^2), so this is the base for the Appell sequence with e.g.f. e^(t^2) e^(x*t) = exp(P(.,x),t) (reverse A059344, cf. A099174, A066325 also). P(n,x) = (a. + x)^n with (a.)^n = a_n and comprise the umbral compositional inverses for e^(-t^2)e^(x*t) = exp(UP(.,x),t), i.e., UP(n,P(.,t)) = x^n = P(n,UP(.,t)), e.g., (P(.,t))^n = P(n,t).
Equals A000407*2 with leading 1 added. (End)
a(n) is also the number of square roots of any permutation in S_{4*n} whose disjoint cycle decomposition consists of 2*n transpositions. - Luis Manuel Rivera Martínez, Mar 04 2015
Self-convolution gives A076729. - Vladimir Reshetnikov, Oct 11 2016
For n > 1, it follows from the formula dated Aug 07 2013 that a(n) is a Zumkeller number (A083207). - Ivan N. Ianakiev, Feb 28 2017
For n divisible by 4, a(n/4) is the number of ways to place n points on an n X n grid with pairwise distinct abscissae, pairwise distinct ordinates, and 90-degree rotational symmetry. For n == 1 (mod 4), the number of ways is a((n-1)/4) because the center point can be considered "fixed". For 180-degree rotational symmetry see A006882, for mirror symmetry see A000085, A135401, and A297708. - Manfred Scheucher, Dec 29 2017

			The following permutations of order 8 and their reversals have this property:
  1 7 3 5 2 4 0 6
  1 7 4 2 5 3 0 6
  2 3 7 6 1 0 4 5
  2 4 7 1 6 0 3 5
  3 2 6 7 0 1 5 4
  3 5 1 7 0 6 2 4

D. E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.2.1.6, Eq. 32.
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N. J. A. Sloane, Table of n, a(n) for n = 0..100
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Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, Australas. J. Combin. 52 (2012), 41-54 (Theorem 1).
Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, arXiv:1005.1531 [math.CO], 2010-2011.
Édouard Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 221.
R. J. Marsh and P. P. Martin, Pascal arrays: counting Catalan sets, arXiv:math/0612572 [math.CO], 2006.
Calin D. Morosan, On the number of broadcast schemes in networks, Information Processing Letters, Volume 100, Issue 5 (2006), 188-193.
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
C. Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.
J. Riordan, Letter to N. J. A. Sloane, Feb 03 1975 (with notes by njas)
H. E. Salzer, Coefficients for expressing the first thirty powers in terms of the Hermite polynomials, Math. Comp., 3 (1948), 167-169.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See pp. 12-13.
Index to divisibility sequences
Index entries for related partition-counting sequences

Cf. A037224, A048854, A001147, A007696, A008545, A122670 (essentially the same sequence), A000165, A047055, A047657, A084947, A084948, A084949, A010050, A000142, A008275, A000108, A000984, A008276, A000680, A094216.
Catalan(n-1)*M^(n-1)*n! for M=1,2,3,4,5,6: A001813, A052714 (or A144828), A221954, A052734, A221953, A221955.
Cf. A123023, A059344, A099174, A066325, A001700, A000407, A006882.
Cf. A000085, A135401, A297708. - Manfred Scheucher, Jan 07 2018

List([0..20],n->Factorial(2*n)/Factorial(n)); # Muniru A Asiru, Nov 01 2018

[Factorial(2*n)/Factorial(n): n in [0..20]]; // Vincenzo Librandi, Oct 09 2018

A001813 := n->(2*n)!/n!;
A001813 := n -> mul(k, k = select(k-> k mod 4 = 2,[$1 .. 4*n])):
seq(A001813(n), n=0..16);  # Peter Luschny, Jun 23 2011

Table[(2n)!/n!, {n,0,20}] (* Harvey P. Dale, May 02 2011 *)

makelist(binomial(n+n, n)*n!,n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=binomial(n+n,n)*n! \\ Charles R Greathouse IV, Jun 15 2011

first(n) = x='x+O('x^n); Vec(serlaplace((1 - 4*x)^(-1/2))) \\ Iain Fox, Jan 01 2018 (corrected by Iain Fox, Jan 11 2018)

from math import factorial
def A001813(n): return factorial(n<<1)//factorial(n) # Chai Wah Wu, Feb 14 2023

[binomial(2*n,n)*factorial(n) for n in range(0, 17)] # Zerinvary Lajos, Dec 03 2009

E.g.f.: (1-4*x)^(-1/2).
a(n) = (2*n)!/n! = Product_{k=0..n-1} (4*k + 2) = A081125(2*n).
Integral representation as n-th moment of a positive function on a positive half-axis: a(n) = Integral_{x=0..oo} x^n*exp(-x/4)/(sqrt(x)*2*sqrt(Pi)) dx, n >= 0. This representation is unique. - Karol A. Penson, Sep 18 2001
Define a'(1)=1, a'(n) = Sum_{k=1..n-1} a'(n-k)*a'(k)*C(n, k); then a(n)=a'(n+1). - Benoit Cloitre, Apr 27 2003
With interpolated zeros (1, 0, 2, 0, 12, ...) this has e.g.f. exp(x^2). - Paul Barry, May 09 2003
a(n) = A000680(n)/A000142(n)*A000079(n) = Product_{i=0..n-1} (4*i + 2) = 4^n*Pochhammer(1/2, n) = 4^n*GAMMA(n+1/2)/sqrt(Pi). - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
For asymptotics, see the Robinson paper.
a(k) = (2*k)!/k! = Sum_{i=1..k+1} |A008275(i,k+1)| * k^(i-1). - André F. Labossière, Jun 21 2007
a(n) = 12*A051618(a) n >= 2. - Zerinvary Lajos, Feb 15 2008
a(n) = A000984(n)*A000142(n). - Zerinvary Lajos, Mar 25 2008
a(n) = A016825(n-1)*a(n-1). - Roger L. Bagula, Sep 17 2008
a(n) = (-1)^n*A097388(n). - D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008
From Paul Barry, Jan 15 2009: (Start)
G.f.: 1/(1-2x/(1-4x/(1-6x/(1-8x/(1-10x/(1-... (continued fraction);
a(n) = (n+1)!*A000108(n). (End)
a(n) = Sum_{k=0..n} A132393(n,k)*2^(2n-k). - Philippe Deléham, Feb 10 2009
G.f.: 1/(1-2x-8x^2/(1-10x-48x^2/(1-18x-120x^2/(1-26x-224x^2/(1-34x-360x^2/(1-42x-528x^2/(1-... (continued fraction). - Paul Barry, Nov 25 2009
a(n) = A173333(2*n,n) for n>0; cf. A006963, A001761. - Reinhard Zumkeller, Feb 19 2010
From Gary W. Adamson, Jul 19 2011: (Start)
a(n) = upper left term of M^n, M = an infinite square production matrix as follows:
  2, 2, 0, 0, 0, 0, ...
  4, 4, 4, 0, 0, 0, ...
  6, 6, 6, 6, 0, 0, ...
  8, 8, 8, 8, 8, 0, ...
  ...
(End)
a(n) = (-2)^n*Sum_{k=0..n} 2^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 1/Q(0), where Q(k) = 1 + x*(4*k+2) - x*(4*k+4)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 18 2013
G.f.: 2/G(0), where G(k) = 1 + 1/(1 - x*(8*k+4)/(x*(8*k+4) - 1 + 8*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 2*x/(2*x + 1/(2*k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
D-finite with recurrence: a(n) = (4*n-6)*a(n-2) + (4*n-3)*a(n-1), n>=2. - Ivan N. Ianakiev, Aug 07 2013
Sum_{n>=0} 1/a(n) = (exp(1/4)*sqrt(Pi)*erf(1/2) + 2)/2 = 1 + A214869, where erf(x) is the error function. - Ilya Gutkovskiy, Nov 10 2016
Sum_{n>=0} (-1)^n/a(n) = 1 - sqrt(Pi)*erfi(1/2)/(2*exp(1/4)), where erfi(x) is the imaginary error function. - Amiram Eldar, Feb 20 2021
a(n) = 1/([x^n] hypergeom([1], [1/2], x/4)). - Peter Luschny, Sep 13 2024
a(n) = 2^n*n!*JacobiP(n, -1/2, -n, 3). - Peter Luschny, Jan 22 2025
G.f.: 2F0(1,1/2;;4x). - R. J. Mathar, Jun 07 2025

More terms from James Sellers, May 01 2000

A001599 Harmonic or Ore numbers: numbers k such that the harmonic mean of the divisors of k is an integer.

Original entry on oeis.org

1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190, 18600, 18620, 27846, 30240, 32760, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 332640, 360360, 539400, 695520, 726180, 753480, 950976, 1089270, 1421280, 1539720

Offset: 1

N. J. A. Sloane

Note that the harmonic mean of the divisors of k = k*tau(k)/sigma(k).
Equivalently, k*tau(k)/sigma(k) is an integer, where tau(k) (A000005) is the number of divisors of k and sigma(k) is the sum of the divisors of k (A000203).
Equivalently, the average of the divisors of k divides k.
Note that the average of the divisors of k is not necessarily an integer, so the above wording should be clarified as follows: k divided by the average is an integer. See A007340. - Thomas Ordowski, Oct 26 2014
Ore showed that every perfect number (A000396) is harmonic. The converse does not hold: 140 is harmonic but not perfect. Ore conjectured that 1 is the only odd harmonic number.
Other examples of power mean numbers k such that some power mean of the divisors of k is an integer are the RMS numbers A140480. - Ctibor O. Zizka, Sep 20 2008
Conjecture: Every harmonic number is practical (A005153). I've verified this refinement of Ore's conjecture for all terms less than 10^14. - Jaycob Coleman, Oct 12 2013
Conjecture: All terms > 1 are Zumkeller numbers (A083207). Verified for all n <= 50. - Ivan N. Ianakiev, Nov 22 2017
Verified for n <= 937. - David A. Corneth, Jun 07 2020
Kanold (1957) proved that the asymptotic density of the harmonic numbers is 0. - Amiram Eldar, Jun 01 2020
Zachariou and Zachariou (1972) called these numbers "Ore numbers", after the Norwegian mathematician Øystein Ore (1899 - 1968), who was the first to study them. Ore (1948) and Garcia (1954) referred to them as "numbers with integral harmonic mean of divisors". The term "harmonic numbers" was used by Pomerance (1973). They are sometimes called "harmonic divisor numbers", or "Ore's harmonic numbers", to differentiate them from the partial sums of the harmonic series. - Amiram Eldar, Dec 04 2020
Conjecture: all terms > 1 have a Mersenne prime as a factor. - Ivan Borysiuk, Jan 28 2024

			k=140 has sigma_0(140)=12 divisors with sigma_1(140)=336. The average divisor is 336/12=28, an integer, and divides k: k=5*28, so 140 is in the sequence.
k=496 has sigma_0(496)=10, sigma_1(496)=992: the average divisor 99.2 is not an integer, but k/(sigma_1/sigma_0)=496/99.2=5 is an integer, so 496 is in the sequence.

G. L. Cohen and Deng Moujie, On a generalization of Ore's harmonic numbers, Nieuw Arch. Wisk. (4), 16 (1998) 161-172.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, Section B2, pp. 74-75.
W. H. Mills, On a conjecture of Ore, Proc. Number Theory Conf., Boulder CO, 1972, 142-146.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.

Robert G. Wilson v, Table of n, a(n) for n = 1..937 (terms n = 1..170 from T. D. Noe and Klaus Brockhaus)
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, The Biharmonic mean, arXiv:1601.03081 [math.NT], 2016.
Abiodun E. Adeyemi, A Study of @-numbers, arXiv:1906.05798 [math.NT], 2019.
Ivan Borysiuk, Conjectured to be the 10000 smallest Ore numbers
Graeme L. Cohen, Email to N. J. A. Sloane, Apr. 1994.
Graeme L. Cohen, Numbers whose positive divisors have small integral harmonic mean, Mathematics of Computation, Vol. 66, No. 218, (1997), pp. 883-891.
Graeme L. Cohen and Ronald M. Sorli, Harmonic seeds, Fibonacci Quart., Vol. 36, No. 5 (1998), pp. 386-390; errata, 39 (2001) 4.
Graeme L. Cohen and Ronald M. Sorli, Odd harmonic numbers exceed 10^24, Math. Comp., Vol. 79, No. 272 (2010), pp. 2451-2460.
Mariano Garcia, On numbers with integral harmonic mean, Amer. Math. Monthly, Vol. 61, No. 2 (1954), pp. 89-96.
Takeshi Goto, All harmonic numbers less than 10^14.
Takeshi Goto, Table of a(n) for n = 1..937.
T. Goto and S. Shibata, All numbers whose positive divisors have integral harmonic mean up to 300, Math. Comput., Vol. 73, No. 245 (2004), pp. 475-491.
Richard K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991.
Hans-Joachim Kanold, Über das harmonische Mittel der Teiler einer natürlichen Zahl, Math. Ann., Vol. 133 (1957), pp. 371-374.
Oystein Ore, On the averages of the divisors of a number, Amer. Math. Monthly, Vol. 55, No. 10 (1948), pp. 615-619.
Oystein Ore, On the averages of the divisors of a number. (annotated scanned copy)
Carl Pomerance, On a Problem of Ore: Harmonic Numbers, unpublished manuscript, 1973; abstract *709-A5, Notices of the American Mathematical Society, Vol. 20, 1973, page A-648, entire volume.
Eric Weisstein's World of Mathematics, Harmonic Mean.
Eric Weisstein's World of Mathematics, Harmonic Divisor Number.
Wikipedia, Harmonic mean.
Wikipedia, Harmonic divisor number.
Andreas and Eleni Zachariou, Perfect, semi-perfect and Ore numbers, Bull. Soc. Math. Grèce (New Ser.), Vol. 13, No. 13A (1972), pp. 12-22; alternative link.
Index entries for sequences where any odd perfect numbers must occur

See A003601 for analogs referring to arithmetic mean and A000290 for geometric mean of divisors.
See A001600 and A090240 for the integer values obtained.
sigma_0(n) (or tau(n)) is the number of divisors of n (A000005).
sigma_1(n) (or sigma(n)) is the sum of the divisors of n (A000203).
Cf. A007340, A090945, A035527, A007691, A074247, A053783. Not a subset of A003601.
Cf. A027750.

Concatenation([1],Filtered([2,4..2000000],n->IsInt(n*Tau(n)/Sigma(n)))); # Muniru A Asiru, Nov 26 2018

import Data.Ratio (denominator)
import Data.List (genericLength)
a001599 n = a001599_list !! (n-1)
a001599_list = filter ((== 1) . denominator . hm) [1..] where
   hm x = genericLength ds * recip (sum $ map (recip . fromIntegral) ds)
          where ds = a027750_row x
-- Reinhard Zumkeller, Jun 04 2013, Jan 20 2012

q:= (p,k) -> p^k*(p-1)*(k+1)/(p^(k+1)-1):
filter:= proc(n) local t; mul(q(op(t)),t=ifactors(n)[2])::integer end proc:
select(filter, [$1..10^6]); # Robert Israel, Jan 14 2016

Do[ If[ IntegerQ[ n*DivisorSigma[0, n]/ DivisorSigma[1, n]], Print[n]], {n, 1, 1550000}]
Select[Range[1600000],IntegerQ[HarmonicMean[Divisors[#]]]&] (* Harvey P. Dale, Oct 20 2012 *)

a(n)=if(n<0,0,n=a(n-1);until(0==(sigma(n,0)*n)%sigma(n,1),n++);n) /* Michael Somos, Feb 06 2004 */

from sympy import divisor_sigma as sigma
def ok(n): return (n*sigma(n, 0))%sigma(n, 1) == 0
print([n for n in range(1, 10**4) if ok(n)]) # Michael S. Branicky, Jan 06 2021

from itertools import count, islice
from functools import reduce
from math import prod
from sympy import factorint
def A001599_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        f = factorint(n)
        s = prod((p**(e+1)-1)//(p-1) for p, e in f.items())
        if not reduce(lambda x,y:x*y%s,(e+1 for e in f.values()),1)*n%s:
            yield n
A001599_list = list(islice(A001599_gen(),20)) # Chai Wah Wu, Feb 14 2023

{ k : A106315(k) = 0 }. - R. J. Mathar, Jan 25 2017

More terms from Klaus Brockhaus, Sep 18 2001

A004394 Superabundant [or super-abundant] numbers: n such that sigma(n)/n > sigma(m)/m for all m < n, sigma(n) being A000203(n), the sum of the divisors of n.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 10080, 15120, 25200, 27720, 55440, 110880, 166320, 277200, 332640, 554400, 665280, 720720, 1441440, 2162160, 3603600, 4324320, 7207200, 8648640, 10810800, 21621600

Offset: 1

Matthew Conroy

Matthew Conroy points out that these are different from the highly composite numbers - see A002182. Jul 10 1996
With respect to the comment above, neither sequence is subsequence of the other. - Ivan N. Ianakiev, Feb 11 2020
Also n such that sigma_{-1}(n) > sigma_{-1}(m) for all m < n, where sigma_{-1}(n) is the sum of the reciprocals of the divisors of n. - Matthew Vandermast, Jun 09 2004
Ramanujan (1997, Section 59; written in 1915) called these numbers "generalized highly composite." Alaoglu and Erdős (1944) changed the terminology to "superabundant." - Jonathan Sondow, Jul 11 2011
Alaoglu and Erdős show that: (1) n is superabundant => n=2^{e_2} * 3^{e_3} * ...* p^{e_p}, with e_2 >= e_3 >= ... >= e_p (and e_p is 1 unless n=4 or n=36); (2) if q < r are primes, then | e_r - floor(e_q*log(q)/log(r)) | <= 1; (3) q^{e_q} < 2^{e_2+2} for primes q, 2 < q <= p. - Keith Briggs, Apr 26 2005
It follows from Alaoglu and Erdős finding 1 (above) that, for n > 7, a(n) is a Zumkeller Number (A083207); for details, see Proposition 9 and Corollary 5 at Rao/Peng link (below). - Ivan N. Ianakiev, Feb 11 2020
See A166735 for superabundant numbers that are not highly composite, and A189228 for superabundant numbers that are not colossally abundant.
Pillai called these numbers "highly abundant numbers of the 1st order". - Amiram Eldar, Jun 30 2019

R. Honsberger, Mathematical Gems, M.A.A., 1973, p. 112.
J. Sandor, "Abundant numbers", In: M. Hazewinkel, Encyclopedia of Mathematics, Supplement III, Kluwer Acad. Publ., 2002 (see pp. 19-21).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 128.

D. Kilminster, Table of n, a(n) for n = 1..2000 (extends to n = 8436 in the comments; first 500 terms from T. D. Noe)
A. Akbary and Z. Friggstad, Superabundant numbers and the Riemann hypothesis, Amer. Math. Monthly, 116 (2009), 273-275.
L. Alaoglu and P. Erdős, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Errata.
Christian Axler, Inequalities involving the primorial counting function, arXiv:2406.04018 [math.NT], 2024. See p. 12.
Yu. Bilu, P. Habegger, and L. Kühne, Effective bounds for singular units, arXiv:1805.07167 [math.NT], 2018.
Benjamin Braun and Brian Davis, Antichain Simplices, arXiv:1901.01417 [math.CO], 2019.
Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 16 (2006), p. 251-256.
Tibor Burdette and Ian Stewart, Counterexamples to a Conjecture by Alaoglu and Erdős, arXiv:2009.03306 [math.NT], 2020.
Geoffrey Caveney, Jean-Louis Nicolas and Jonathan Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, INTEGERS 11 (2011), #A33.
Geoffrey Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv preprint arXiv:1112.6010 [math.NT], 2011. - From _N. J. A. Sloane_, Apr 14 2012
Geoffrey Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, Ramanujan J., 29 (2012), 359-384.
François Clément and Stefan Steinerberger, On the largest singular vector of the Redheffer matrix, arXiv:2502.09489 [math.NT], 2025. See p. 2.
P. Erdős and J.-L. Nicolas, Répartition des nombres superabondants (Text in French), Bulletin de la S. M. F., tome 103 (1975), pp. 65-90.
F. Jokar, On k-layered numbers and some labeling related to k-layered numbers, arXiv:2003.11309 [math.NT], 2020.
Stepan Kochemazov, Oleg Zaikin, Eduard Vatutin, and Alexey Belyshev, Enumerating Diagonal Latin Squares of Order Up to 9, J. Int. Seq., Vol. 23 (2020), Article 20.1.2.
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543.
A. Morkotun, On the increase of Gronwall function value at the multiplication of its argument by a prime, arXiv preprint arXiv:1307.0083 [math.NT], 2013.
S. Nazardonyavi and S. Yakubovich, Superabundant numbers, their subsequences and the Riemann hypothesis, arXiv preprint arXiv:1211.2147 [math.NT], 2012.
S. Nazardonyavi and S. Yakubovich, Delicacy of the Riemann hypothesis and certain subsequences of superabundant numbers, arXiv preprint arXiv:1306.3434 [math.NT], 2013.
S. Nazardonyavi and S. Yakubovich, Extremely Abundant Numbers and the Riemann Hypothesis, Journal of Integer Sequences, 17 (2014), Article 14.2.8.
Walter Nissen, Abundancy : Some Resources.
T. D. Noe, First 500 superabundant numbers.
T. D. Noe, First 1000000 superabundant numbers (21 MB, zipped).
S. Sivasankaranarayana Pillai, Highly abundant numbers, Bulletin of the Calcutta Mathematical Society, Vol. 35, No. 1 (1943), pp. 141-156.
S. Sivasankaranarayana Pillai, On numbers analogous to highly composite numbers of Ramanujan, Rajah Sir Annamalai Chettiar Commemoration Volume, ed. Dr. B. V. Narayanaswamy Naidu, Annamalai University, 1941, pp. 697-704.
S. Ramanujan, Highly composite numbers, Annotated and with a foreword by J.-L. Nicolas and G. Robin, Ramanujan J., 1 (1997), 119-153.
K. P. S. Bhaskara Rao and Yuejian Peng, On Zumkeller Numbers, Journal of Number Theory, Volume 133, Issue 4, April 2013, pp. 1135-1155.
T. Schwabhäuser, Preventing Exceptions to Robin's Inequality, arXiv preprint arXiv:1308.3678 [math.NT], 2013.
Eric Weisstein's World of Mathematics, Superabundant Number.
Wikipedia, Superabundant number.

Almost the same as A077006.
The colossally abundant numbers A004490 are a subsequence, as are A023199.
Subsequence of A025487; apart from a(3) = 4 and a(7) = 36, a subsequence of A102750.
Cf. A000203, A002093, A002182.
Cf. A112974 (number of superabundant numbers between colossally abundant numbers).
Cf. A091901 (Robin's inequality), A189686 (superabundant and the reverse of Robin's inequality), A192884 (non-superabundant and the reverse of Robin's inequality).

a=0; Do[b=DivisorSigma[1, n]/n; If[b>a, a=b; Print[n]], {n, 1, 10^7}]
(* Second program: convert all 8436 terms in b-file into a list of terms: *)
f[w_] := Times @@ Flatten@ {Complement[#1, Union[#2, #3]], Product[Prime@ i, {i, PrimePi@ #}] & /@ #2, Factorial /@ #3} & @@ ToExpression@ {StringSplit[w, ?(! DigitQ@ # &)], StringCases[w, (x : DigitCharacter ..) ~~ "#" :> x], StringCases[w, (x : DigitCharacter ..) ~~ "!" :> x]}; Map[Which[StringTake[#, 1] == {"#"}, f@ Last@ StringSplit@ Last@ #, StringTake[#, 1] == {}, Nothing, True, ToExpression@ StringSplit[#][[1, -1]]] &, Drop[Import["b004394.txt", "Data"], 3] ] (* _Michael De Vlieger, May 08 2018 *)

print1(r=1);forstep(n=2,1e6,2,t=sigma(n,-1);if(t>r,r=t;print1(", "n))) \\ Charles R Greathouse IV, Jul 19 2011

a(n+1) <= 2*a(n). - A.H.M. Smeets, Jul 10 2021

Name edited by Peter Munn, Mar 13 2019

A028983 Numbers whose sum of divisors is even.

Original entry on oeis.org

3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 82

Offset: 1

Patrick De Geest

The even terms of this sequence are the even terms appearing in A178910. [Edited by M. F. Hasler, Oct 02 2014]
A071324(a(n)) is even. - Reinhard Zumkeller, Jul 03 2008
Sigma(a(n)) = A000203(a(n)) = A152678(n). - Jaroslav Krizek, Oct 06 2009
A083207 is a subsequence. - Reinhard Zumkeller, Jul 19 2010
Numbers k such that the number of odd divisors of k (A001227) is even. - Omar E. Pol, Apr 04 2016
Numbers k such that the sum of odd divisors of k (A000593) is even. - Omar E. Pol, Jul 05 2016
Numbers with a squarefree part greater than 2. - Peter Munn, Apr 26 2020
Equivalently, numbers whose odd part is nonsquare. Compare with the numbers whose square part is even (i.e., nonodd): these are the positive multiples of 4, A008586\{0}, and A225546 provides a self-inverse bijection between the two sets. - Peter Munn, Jul 19 2020
Also numbers whose reversed prime indices have alternating product > 1, where we define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). Also Heinz numbers of the partitions counted by A347448. - Gus Wiseman, Oct 29 2021
Numbers whose number of middle divisors is not odd (cf. A067742). - Omar E. Pol, Aug 02 2022

T. D. Noe, Table of n, a(n) for n = 1..1000

The complement is A028982 = A000290 U A001105.
Cf. A000203, A000593, A001227, A007913, A178910, A152678, A067742.
Subsequences: A083207, A091067, A145204\{0}, A225838, A225858.
Cf. A334748 (a permutation).
Related to A008586 via A225546.
Ranks the partitions counted by A347448, complement A119620.
Cf. A030059, A335433, A335448, A339890, A344607, A347438, A347443, A347445, A347446, A347452, A347453, A347465.

Select[Range[82],EvenQ[DivisorSigma[1,#]]&] (* Jayanta Basu, Jun 05 2013 *)

is(n)=!issquare(n)&&!issquare(n/2) \\ Charles R Greathouse IV, Jan 11 2013

from math import isqrt
def A028983(n):
    def f(x): return n-1+isqrt(x)+isqrt(x>>1)
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 22 2024

a(n) ~ n. - Charles R Greathouse IV, Jan 11 2013
a(n) = n + (1 + sqrt(2)/2)*sqrt(n) + O(1). - Charles R Greathouse IV, Sep 01 2015
A007913(a(n)) > 2. - Peter Munn, May 05 2020

A002827 Unitary perfect numbers: numbers k such that usigma(k) - k = k.

Original entry on oeis.org

6, 60, 90, 87360, 146361946186458562560000

Offset: 1

N. J. A. Sloane

d is a unitary divisor of k if gcd(d,k/d)=1; usigma(k) is their sum (A034448).
The prime factors of a unitary perfect number (A002827) are the Higgs primes (A057447). - Paul Muljadi, Oct 10 2005
It is not known if a(6) exists. - N. J. A. Sloane, Jul 27 2015
Frei proved that if there is a unitary perfect number that is not divisible by 3, then it is divisible by 2^m with m >= 144, it has at least 144 distinct odd prime factors, and it is larger than 10^440. - Amiram Eldar, Mar 05 2019
Conjecture: Subsequence of A083207 (Zumkeller numbers). Verified for all present terms. - Ivan N. Ianakiev, Jan 20 2020

			Unitary divisors of 60 are 1,4,3,5,12,20,15,60, with sum 120 = 2*60.
146361946186458562560000 = 2^18 * 3 * 5^4 * 7 * 11 * 13 * 19 * 37 * 79 * 109 * 157 * 313.

R. K. Guy, Unsolved Problems in Number Theory, Sect. B3.
F. Le Lionnais, Les Nombres Remarquables. Paris: Hermann, p. 59, 1983.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.45.1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 147-148.

H. A. M. Frei, Über unitar perfekte Zahlen, Elemente der Mathematik, Vol. 33, No. 4 (1978), pp. 95-96.
Takeshi Goto, Upper Bounds for Unitary Perfect Numbers and Unitary Harmonic Numbers, Rocky Mountain Journal of Mathematics, Vol. 37, No. 5 (2007), pp. 1557-1576.
A. V. Lelechenko, The Quest for the Generalized Perfect Numbers, in Theoretical and Applied Aspects of Cybernetics, TAAC 2014, Kiev.
M. V. Subbarao, Letter to N. J. A. Sloane, Feb 18 1974
M. V. Subbarao, T. J. Cook, R. S. Newberry and J. M. Weber, On unitary perfect numbers, Delta, 3 (No. 1, 1972), 22-26.
G. Villemin's Almanac of Numbers, Nombres Unitairement Parfaits
C. R. Wall, Letter to P. Hagis, Jr., Jan 13 1972
C. R. Wall, The fifth unitary perfect number, Canad. Math. Bull., 18 (1975), 115-122.
C. R. Wall, On the largest odd component of a unitary perfect number, Fib. Quart., 25 (1987), 312-316.
Eric Weisstein's World of Mathematics, Unitary Perfect Number.
Wikipedia, Unitary perfect number

Cf. A034460, A034448, A057447.
Subsequence of the following sequences: A003062, A290466 (seemingly), A293188, A327157, A327158.
Gives the positions of ones in A327159.

usnQ[n_]:=Total[Select[Divisors[n],GCD[#,n/#]==1&]]==2n; Select[Range[ 90000],usnQ] (* This will generate the first four terms of the sequence; it would take a very long time to attempt to generate the fifth term. *) (* Harvey P. Dale, Nov 14 2012 *)

is(n)=sumdivmult(n, d, if(gcd(d, n/d)==1, d))==2*n \\ Charles R Greathouse IV, Aug 01 2016

If m is a term and omega(m) = A001221(m) = k, then m < 2^(2^k) (Goto, 2007). - Amiram Eldar, Jun 06 2020

A111592 Admirable numbers. A number n is admirable if there exists a proper divisor d' of n such that sigma(n)-2d'=2n, where sigma(n) is the sum of all divisors of n.

Original entry on oeis.org

12, 20, 24, 30, 40, 42, 54, 56, 66, 70, 78, 84, 88, 102, 104, 114, 120, 138, 140, 174, 186, 222, 224, 234, 246, 258, 270, 282, 308, 318, 354, 364, 366, 368, 402, 426, 438, 464, 474, 476, 498, 532, 534, 582, 606, 618, 642, 644, 650, 654, 672, 678, 762, 786, 812

Offset: 1

Jason Earls, Aug 09 2005

All admirable numbers are abundant.
If 2^n-2^k-1 is an odd prime then m=2^(n-1)*(2^n-2^k-1) is in the sequence because 2^k is one of the proper divisors of m and sigma(m)-2m=(2^n-1)*(2^n-2^k)-2^n*(2^n-2^k-1)=2^k hence m=(sigma(m)-m)-2^k, namely m is an Admirable number. This is one of the results of the following theorem that I have found. Theorem: If 2^n-j-1 is an odd prime and m=2^(n-1)*(2^n-j-1) then sigma(m)-2m=j. The case j=0 is well known. - Farideh Firoozbakht, Jan 28 2006
In particular, these numbers have abundancy 2 to 3: 2 < sigma(n)/n <= 3. - Charles R Greathouse IV, Jan 30 2014
Subsequence of A083207. - Ivan N. Ianakiev, Mar 20 2017
The concept of admirable numbers was developed by educator Jerome Michael Sachs (1914-2012) for a television in-service training course in mathematics for elementary school teachers. - Amiram Eldar, Aug 22 2018
Odd terms are listed in A109729. For abundant nonsquares, it is equivalent to say sigma(n)/2 - n divides n. For squares, sigma(n)/2 - n is half-integer, but n could still be an integer multiple. This first occurs for n = m^2 with even m = 2^k*(2^(2*k+1)-1), k = 1, 2, 3, 6, ... (A146768), and odd m = 13167. - M. F. Hasler, Jan 26 2020

			12 = 1+3+4+6-2, 20 = 2+4+5+10-1, etc.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Giovanni Resta, admirable numbers
J. M. Sachs, Admirable Numbers and Compatible Pairs, The Arithmetic Teacher, Vol. 7, No. 6 (1960), pp. 293-295.
T. Trotter, Admirable Numbers

Subsequence of A005101 (abundant numbers).
Cf. A000396 (perfect numbers), A005100 (deficient numbers), A000203 (sigma), A061645.
Cf. A109729 (odd admirable numbers).

with(numtheory); isadmirable := proc(n) local b, d, S; b:=false; S:=divisors(n) minus {n}; for d in S do if sigma(n)-2*d=2*n then b:=true; break fi od; return b; end: select(proc(z) isadmirable(z) end, [$1..1000]); # Walter Kehowski, Aug 12 2005

fQ[n_] := Block[{d = Most[Divisors[n]], k = 1}, l = Length[d]; s = Plus @@ d; While[k < l && s - 2d[[k]] > n, k++ ]; If[k > l || s != n + 2d[[k]], False, True]]; Select[ Range[821], fQ[ # ] &] (* Robert G. Wilson v, Aug 13 2005 *)
Select[Range[812],MemberQ[Most[Divisors[#]],(DivisorSigma[1,#]-2*#)/2]&] (* Ivan N. Ianakiev, Mar 23 2017 *)

for(n=1,10^3,ap=sigma(n)-2*n;if(ap>0 && (ap%2)==0,d=ap/2;if(d!=n && (n%d)==0, print1(n",")))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 30 2008

is(n)=if(issquare(n)||issquare(n/2),0,my(d=sigma(n)/2-n); d>0 && d!=n && n%d==0) \\ Charles R Greathouse IV, Jun 21 2011

Better definition from Walter Kehowski, Aug 12 2005

A083206 a(n) is the number of ways of partitioning the divisors of n into two disjoint sets with equal sum.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 17, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 3, 0, 0, 0, 14, 0, 0, 0, 1, 0, 13, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 2, 0, 1

Offset: 1

Reinhard Zumkeller, Apr 22 2003

nonn

a(n)=0 for deficient numbers n (A005100), but the converse is not true, as 18 is abundant (A005101) and a(18)=0, see A083211;
a(n)=1 for perfect numbers n (A000396), see A083209 for all numbers with a(n)=1;
records: A083213(k)=a(A083212(k)).
In order that a(n)>0, the sum of divisors of n must be even by definition: a(n) = half the number of partitions of A000203(n)/2 into divisors of n, see formula. [Reinhard Zumkeller, Jul 10 2010]

			a(24)=3: 1+2+3+4+8+12=6+24, 1+3+6+8+12=2+4+24, 4+6+8+12=1+2+3+24.

Antti Karttunen, Table of n, a(n) for n = 1..101632 (terms 1..800 from Reinhard Zumkeller, terms 801..10000 from T. D. Noe).
Reinhard Zumkeller, Illustration of initial terms

Cf. A083208 [= a(A083207(n))], A083211, A000005, A000203, A082729, A378446 (inverse Möbius transform), A378449.
Cf. A083207 (positions of terms > 0), A083210 (positions of 0's), A083209 (positions of 1's), A378652 (of 2's).
Cf. also A033630, A065205, A058377, A325806.

a[n_] := (s = DivisorSigma[1, n]; If[Mod[s, 2] == 1, 0, f[n, s/2, 2]]); f[n_, m_, k_] := f[n, m, k] = If[k <= m, f[n, m, k+1] + f[n, m-k, k+1]*Boole[Mod[n, k] == 0], Boole[m == 0]]; Array[a, 105] (* Jean-François Alcover, Jul 29 2015, after Reinhard Zumkeller *)

A083206(n) = { my(s=sigma(n),p=1); if(s%2 || s < 2*n, 0, fordiv(n, d, p *= ('x^d + 'x^-d)); (polcoeff(p, 0)/2)); }; \\ Antti Karttunen, Dec 02 2024, after Ilya Gutkovskiy

a(n) = if sigma(n) mod 2 = 1 then 0 else f(n,sigma(n)/2,2), where sigma=A000203 and f(n,m,k) = if k<=m then f(n,m,k+1)+f(n,m-k,k+1)*0^(n mod k) else 0^m, cf. A033630, also using f. [Reinhard Zumkeller, Jul 10 2010]

a(n) is half the coefficient of x^0 in Product_{d|n} (x^d + 1/x^d). - Ilya Gutkovskiy, Feb 04 2024

A036691 Compositorial numbers: product of first n composite numbers.

Original entry on oeis.org

1, 4, 24, 192, 1728, 17280, 207360, 2903040, 43545600, 696729600, 12541132800, 250822656000, 5267275776000, 115880067072000, 2781121609728000, 69528040243200000, 1807729046323200000, 48808684250726400000, 1366643159020339200000

Offset: 0

Felice Russo

a(A196415(n)) = A141092(n) * A053767(A196415(n)). - Reinhard Zumkeller, Oct 03 2011
For n>11, A000142(n) < a(n) < A002110(n). - Chayim Lowen, Aug 18 2015
For n = {2,3,4}, a(n) is testably a Zumkeller number (A083207). For n > 4, a(n) is of the form 2^e_1 * p_2^e_2 * … * p_m^e_m, where e_m = 1 and e = floor(log_2(p_m)) < e_1. Therefore, 2^e * p_m^e_m is primitive Zumkeler number (A180332). Therefore, 2^e_1 * p_m^e_m is a Zumkeller number. Therefore, a(n) = 2^e_1 * p_m^e_m * r, where r is relatively prime to 2*p_m is a Zumkeller number. Therefore, for n > 1, a(n) is a Zumkeller number (see my proof at A002182 for details). - Ivan N. Ianakiev, May 04 2020

			a(3) = c(1)*c(2)*c(3) = 4*6*8 = 192.

T. D. Noe, Table of n, a(n) for n = 0..100
Googology Wiki, Compositorial

Cf. primorial numbers A002110. Distinct members of A049614. See also A049650, A060880.

Cf. A092435 (subsequence: A092435(n) = a(prime(n)-n-1)). - Chayim Lowen, Jul 23 2015

a036691_list = scanl1 (*) a002808_list -- Reinhard Zumkeller, Oct 03 2011

A036691 := proc(n)
        mul(A002808(i),i=1..n) ;
end proc: # R. J. Mathar, Oct 03 2011

Composite[n_] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; Table[ Product[ Composite[i], {i, 1, n}], {n, 0, 18}] (* Robert G. Wilson v, Sep 13 2003 *)
nn=50;cnos=Complement[Range[nn],Prime[Range[PrimePi[nn]]]];Rest[FoldList[ Times,1,cnos]] (* Harvey P. Dale, May 19 2011 *)
A036691 = Union[Table[n!/(Times@@Prime[Range[PrimePi[n]]]), {n, 29}]] (* Alonso del Arte, Sep 21 2011 *)
Join[{1},FoldList[Times,Select[Range[30],CompositeQ]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 14 2019 *)

a(n)=my(c,p);c=4;p=1;while(n>0,if(!isprime(c),p=p*c;n=n-1);c=c+1);p \\ Ralf Stephan, Dec 21 2013

from sympy import factorial, primepi, primorial, composite
def A036691(n):
    return factorial(composite(n))//primorial(primepi(composite(n))) if n > 0 else 1 # Chai Wah Wu, Sep 08 2020

From Chayim Lowen, Jul 23 - Aug 05 2015: (Start)
a(n) = A049614(A002808(n)) = A000142(A002808(n))/A034386(A002808(n)).
a(n) = Product_{k=1..A002808(n)-n-1} prime(k)^(A085604(A002808(n),k)-1).
Sum_{k >= 1} 1/a(k) = 1.2975167655550616507663335821769... is to this sequence as e is to the factorials. (End)

Corrected and extended by Niklas Eriksen (f95-ner(AT)nada.kth.se) and N. J. A. Sloane

A001034 Orders of noncyclic simple groups (without repetition).

Original entry on oeis.org

60, 168, 360, 504, 660, 1092, 2448, 2520, 3420, 4080, 5616, 6048, 6072, 7800, 7920, 9828, 12180, 14880, 20160, 25308, 25920, 29120, 32736, 34440, 39732, 51888, 58800, 62400, 74412, 95040, 102660, 113460, 126000, 150348, 175560, 178920

Offset: 1

N. J. A. Sloane, Simon Plouffe

An alternative definition, to assist in searching: Orders of non-cyclic finite simple groups.
This comment is about the three sequences A001034, A060793, A056866: The Feit-Thompson theorem says that a finite group with odd order is solvable, hence all numbers in this sequence are even. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 08 2001 [Corrected by Isaac Saffold, Aug 09 2021]
The primitive elements are A257146. These are also the primitive elements of A056866. - Charles R Greathouse IV, Jan 19 2017
Conjecture: This is a subsequence of A083207 (Zumkeller numbers). Verified for n <= 156. A fast provisional test was used, based on Proposition 17 from Rao/Peng JNT paper (see A083207). For those few cases where the fast test failed (such as 2588772 and 11332452) the comprehensive (but much slower) test by T. D. Noe at A083207 was used for result confirmation. - Ivan N. Ianakiev, Jan 11 2020
From M. Farrokhi D. G., Aug 11 2020: (Start)
The conjecture is not true. The smallest and the only counterexample among the first 457 terms of the sequence is a(175) = 138297600.
On the other hand, the orders of sporadic simple groups are Zumkeller. And with the exception of the smallest two orders 7920 and 95040, the odd part of the other orders are also Zumkeller. (End)
Every term in this sequence is divisible by 4*p*q, where p and q are distinct odd primes. - Isaac Saffold, Oct 24 2021

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].
Dickson L.E. Linear groups, with an exposition of the Galois field theory (Teubner, 1901), p. 309.
M. Hall, Jr., A search for simple groups of order less than one million, pp. 137-168 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. Farrokhi D. G., Table of n, a(n) for n = 1..10000
C. Cato, The orders of the known simple groups as far as one trillion, Math. Comp., 31 (1977), 574-577.
L. E. Dickson, Linear Groups with an Exposition of the Galois Field Theory (page images), Dover, NY, 1958, p. 309.
M. Farrokhi D. G., A GAP function generating the smallest non-cyclic finite simple group order greater than a given number m.
Walter Feit and J. G. Thompson, A solvability criterion for finite groups and some consequences, Proc. N. A. S. 48 (6) (1962) 968.
M. Hall Jr., Simple groups of order less than one million, J. Alg. 20 (1) (1972) 98-102
David A. Madore, More terms
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.
Index entries for sequences related to groups
Index entries for "core" sequences

Cf. A000001, A000679, A005180, A001228, A060793, A056866, A056868, A119630.

Cf. A109379 (orders with repetition), A119648 (orders that are repeated).

cp's OEIS Frontend

A001710 Order of alternating group A_n, or number of even permutations of n letters.

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A001813 Quadruple factorial numbers: a(n) = (2n)!/n!.

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A001599 Harmonic or Ore numbers: numbers k such that the harmonic mean of the divisors of k is an integer.

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A004394 Superabundant [or super-abundant] numbers: n such that sigma(n)/n > sigma(m)/m for all m < n, sigma(n) being A000203(n), the sum of the divisors of n.

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A028983 Numbers whose sum of divisors is even.

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