A007489 -id:A007489 - cp's OEIS frontend

A180632 Minimum length of a string over the alphabet A = {1,2,...,n} that contains every permutation of A as a substring exactly once, also known as length of the minimal super-permutation.

0, 1, 3, 9, 33, 153

Offset: 0

Michael Hamm, Sep 13 2010

Obviously bounded below by n! + n - 1 and above by 2(n! - (n - 1)!) + 1.
A better lower bound is n! + (n - 1)! + (n - 2)! + n - 3 (A376269) and a better upper bound is A007489(n). - Nathaniel Johnston, Apr 22 2013
The above mentioned lower bound was essentially shown in 2011 by an anonymous poster on the internet and, filling in some minor details, brought into a formal form by Houston, Pantone and Vatter (see reference). - Peter Luschny, Oct 27 2018
Was conjectured to be equal to A007489, but it is now known that a(n) < A007489(n) for all n > 5. - Robin Houston, Aug 22 2014
Different from the minimal supersequence, in which each permutations of n letters can appear as a subsequence instead of a sub-string (i.e., with noncontiguous characters). Refer to A062714. - Maurizio De Leo, Mar 02 2015
In October 2018 Greg Egan found new records for n=7, 8, 9: a(7) <= 5908, a(8) <= 46205, and a(9) <= 408966. More generally, for any n >= 7, a(n) <= n! + (n-1)! + (n-2)! + (n-3)! + n - 3. - Peter Luschny, Oct 26 2018; corrected by Max Alekseyev, Jan 07 2019
In February 2019, Bogdan Coanda found an example showing that a(7) <= 5907. Later the same month, Greg Egan found an example showing a(7) <= 5906. - Robin Houston, Mar 11 2019
a(6) <= 872 = A007489(6) - 1 [Houston 2014]. - M. F. Hasler, Jul 28 2020

			For n = 1, 2, 3, 4, the (unique, up to relabeling the symbols) minimal words are:
1
121
123121321
123412314231243121342132413214321
For n = 5, there are exactly 8 distinct (up to relabeling the symbols) minimal words.
Comment from _N. J. A. Sloane_, Mar 27 2015: From the Houston (2014 arXiv) paper, a superpermutation of length 872 (not known to be minimal, but shorter than the old upper bound of 873):
  1234561234516234512634512364513264513624513642513645213645123465123415 6234152634152364152346152341652341256341253641253461253416253412653412 3564123546123541623541263541236541326543126453162435162431562431652431 6254316245316425314625314265314256314253614253164523146523145623145263 1452361452316453216453126435126431526431256432156423154623154263154236 1542316542315642135642153624153621453621543621534621354621345621346521 3462513462153642156342165342163542163452163425163421564325164325614325 6413256431265432165432615342613542613452613425613426513426153246513246 5312463512463152463125463215463251463254163254613254631245632145632415 6324516324561324563124653214653241653246153264153261453261543265143625 1436521435621435261435216435214635214365124361524361254361245361243561 2436514235614235164235146235142635142365143265413625413652413562413526 41352461352416352413654213654123

D. Ashlock and J. Tillotson. Construction of small superpermutations and minimal injective superstrings. Congressus Numerantium, 93 (1993), 91-98.

Dhruv Ajmera, An O(n) Space Construction of Superpermutations, arXiv:2505.09628 [cs.DM], 2025. See p. 29.
Anonymous 4chan Poster, Robin Houston, Jay Pantone and Vince Vatter, A lower bound on the length of the shortest superpattern, October 25, 2018.
Jeffrey A. Barnett, Permutation Strings
Greg Egan, Superpermutations, October 20, 2018.
Michael Engen and Vincent Vatter, Containing all permutations, Amer. Math. Monthly, 128 (2021), 4-24, section 2; arXiv preprint, arXiv:1810.08252 [math.CO], 2018-2020.
James Grime and Brady Haran, Superpermutations, Numberphile video, 2018.
James Grime, Matt Parker and Robin Houston, New Superpermutations Discovered!, standupmaths video, March 11, 2019.
Robin Houston, Tackling the Minimal Superpermutation Problem, arXiv:1408.5108 [math.CO], 2014.
Robin Houston and Matt Parker, Superpermutations: the maths problem solved by 4chan, standupmaths video (2019).
Nathaniel Johnston, Non-uniqueness of minimal superpermutations, Discrete Math., 313 (2013), 1553-1557.
Nathaniel Johnston, The minimal superpermutation problem, 2013.
Nathaniel Johnston, All minimal superpermutations on five symbols have been found
Miryana Stefanović, Supermutacije (Supermutations), Master's Thesis, Univ. Belgrade (Serbia 2023). See p. 9. (In Serbian)
Wikipedia, Superpermutation
Aaron Williams, Hamiltonicity of the Cayley digraph on the symmetric group generated by sigma = (1 2 ... n) and tau = (1 2), arXiv:1307.2549v3 [math.CO], 2017.

Cf. A188428, A224986, A062714, A007489, A376269.

Row lengths of A332089 (for n >= 1).

Edited and expanded by Nathaniel Johnston, Apr 22 2013
a(5) computed by Benjamin Chaffin and verified by Nathaniel Johnston, Aug 13 2014
Definition edited by Maurizio De Leo, Mar 02 2015; and by Max Alekseyev, Jan 07 2019

A336415 Number of divisors of n! with equal prime multiplicities.

Original entry on oeis.org

1, 1, 2, 4, 6, 10, 13, 21, 24, 28, 33, 49, 53, 85, 94, 100, 104, 168, 173, 301, 307, 317, 334, 590, 595, 603, 636, 642, 652, 1164, 1171, 2195, 2200, 2218, 2283, 2295, 2301, 4349, 4478, 4512, 4519, 8615, 8626, 16818, 16836, 16844, 17101, 33485, 33491, 33507, 33516, 33582

Offset: 0

Gus Wiseman, Jul 22 2020

nonn

A number k has "equal prime multiplicities" (or is "uniform") iff its prime signature is constant, meaning that k is a power of a squarefree number.

			The a(n) uniform divisors of n for n = 1, 2, 6, 8, 30, 36 are the columns:
  1  2  6  8  30  36
     1  3  6  15  30
        2  4  10  16
        1  3   8  15
           2   6  10
           1   5   9
               4   8
               3   6
               2   5
               1   4
                   3
                   2
                   1
In 20!, the multiplicity of the third prime (5) is 4 but the multiplicity of the fourth prime (7) is 2. Hence there are 2^3 - 1 = 3 divisors with all exponents 3 (we subtract |{1}| = 1 from that count as 1 has no exponent 3). - _David A. Corneth_, Jul 27 2020

David A. Corneth, Table of n, a(n) for n = 0..9999
Jon Maiga, Computer-generated formulas for A336415, Sequence Machine.

The version for distinct prime multiplicities is A336414.
The version for nonprime perfect powers is A336416.
Uniform partitions are counted by A047966.
Uniform numbers are A072774, with nonprime terms A182853.
Numbers with distinct prime multiplicities are A130091.
Divisors with distinct prime multiplicities are counted by A181796.
Maximum divisor with distinct prime multiplicities is A327498.
Uniform divisors are counted by A327527.
Maximum uniform divisor is A336618.
1st differences are given by A048675.
Cf. A000005, A000961, A001222, A001597, A007916, A112798, A124010, A327499.
Factorial numbers: A000142, A007489, A022559, A027423, A048656, A071626, A108731, A325272, A325273, A325617.

Table[Length[Select[Divisors[n!],SameQ@@Last/@FactorInteger[#]&]],{n,0,15}]

a(n) = sumdiv(n!, d, my(ex=factor(d)[,2]); (#ex==0) || (vecmin(ex) == vecmax(ex))); \\ Michel Marcus, Jul 24 2020

a(n) = {if(n<2, return(1)); my(f = primes(primepi(n)), res = 1, t = #f); f = vector(#f, i, val(n, f[i])); for(i = 1, f[1], while(f[t] < i, t--; ); res+=(1<David A. Corneth, Jul 27 2020

a(n) = A327527(n!).

Terms a(31) and onwards from David A. Corneth, Jul 27 2020

A227157 Numbers k whose factorial base representation A007623(k) does not contain any nonleading zeros.

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 17, 21, 23, 33, 35, 39, 41, 45, 47, 57, 59, 63, 65, 69, 71, 81, 83, 87, 89, 93, 95, 105, 107, 111, 113, 117, 119, 153, 155, 159, 161, 165, 167, 177, 179, 183, 185, 189, 191, 201, 203, 207, 209, 213, 215, 225, 227, 231, 233, 237, 239, 273

Offset: 1

Antti Karttunen, Jul 04 2013

a(A003422(n)) = A007489(n).
a(A007489(n)) = (n+1)!-1 thus A007489(n) gives the number of terms less than (n+1)! in this sequence.
Equivalently, there are n! terms in the sequence with their magnitude in range n!..(n+1)!.
Also numbers k such that A304036(k) = 1 for k > 0. - Seiichi Manyama, May 06 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..5913 from Antti Karttunen)

The sequence gives all n for which A208575(n) is not zero. Complement of A227187. Subsets: A071156 (apart from zero), A231716, A231720.

Cf. also A003422, A007489, A007623, A153880, A227130, A227132, A304036.

q[n_] := Module[{k = n, m = 2, c = 0, r}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r == 0, c++]; m++]; c == 0]; Select[Range[300], q] (* Amiram Eldar, Jan 23 2024 *)

A084556 n occurs n! times.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 0

Antti Karttunen, Jun 02 2003

nonn

Also minimum i such that A007489(i) >= n.

For n>=1, a(n) gives the length of the n-th permutation in the sequences like A030298 & A030496.

A. Karttunen, Table of n, a(n) for n = 0..46234
Index entries for sequences related to permutations

First differences of A084555. Used to compute A084557. Differs from A084506 first time at the 130th term, where A084506(130) = 6, while A084556(130) = 5. Cf. also A002024, A072643, A072649, A090529.

Flatten[ Table[#, {#!}] & /@ Range[0, 5]]

A335407 Number of anti-run permutations of the prime indices of n!.

Original entry on oeis.org

1, 1, 1, 2, 0, 2, 3, 54, 0, 30, 105, 6090, 1512, 133056, 816480, 127209600, 0, 10090080, 562161600, 69864795000, 49989139200, 29593652088000, 382147120555200, 41810689605484800, 4359985823793600, 3025062801079038720, 49052072750637116160, 25835971971637227375360

Offset: 0

Gus Wiseman, Jul 01 2020

nonn

An anti-run is a sequence with no adjacent equal parts.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Conjecture: Only vanishes at n = 4 and n = 8.
a(16) = 0. Proof: 16! = 2^15 * m where bigomega(m) = A001222(m) = 13. We can't separate 15 1's with 13 other numbers. - David A. Corneth, Jul 04 2020

			The a(0) = 1 through a(6) = 3 anti-run permutations:
  ()  ()  (1)  (1,2)  .  (1,2,1,3,1)  (1,2,1,2,1,3,1)
               (2,1)     (1,3,1,2,1)  (1,2,1,3,1,2,1)
                                      (1,3,1,2,1,2,1)

Andrew Howroyd, Table of n, a(n) for n = 0..200

The version for Mersenne numbers is A335432.
Anti-run compositions are A003242.
Anti-run patterns are counted by A005649.
Permutations of prime indices are A008480.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Strict permutations of prime indices are A335489.
Cf. A056239, A106351, A112798, A114938, A278990, A292884, A325535, A335125.
Factorial numbers: A000142, A001222, A002982, A007489, A022559, A027423, A054991, A108731, A181819, A181821, A325272, A325273, A325617.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n!]],!MatchQ[#,{_,x_,x_,_}]&]],{n,0,10}]

\\ See A335452 for count.
a(n)={count(factor(n!)[,2])} \\ Andrew Howroyd, Feb 03 2021

a(n) = A335452(A000142(n)). - Andrew Howroyd, Feb 03 2021

Terms a(14) and beyond from Andrew Howroyd, Feb 03 2021

A014288 a(n) = floor(Sum_{k=0..n} k!/2), or floor( A003422(n+1)/2 ).

Original entry on oeis.org

0, 1, 2, 5, 17, 77, 437, 2957, 23117, 204557, 2018957, 21977357, 261478157, 3374988557, 46964134157, 700801318157, 11162196262157, 189005910310157, 3390192763174157, 64212742967590157, 1280663747055910157, 26826134832910630157, 588826498721714470157

Offset: 0

N. J. A. Sloane

nonn

The first term a(0) would be a fraction if the floor( ... ) function were omitted; for n >= 2, all terms from A003422 are even. - M. F. Hasler, Dec 16 2007

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A003422, A007489, A067078.

[Floor((&+[Factorial(j): j in [0..n]])/2): n in [0..30]]; // G. C. Greubel, Sep 05 2022

a:= proc(n) a(n):= `if`(n<3, n, (n+1)*a(n-1)-n*a(n-2)) end:
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 01 2013

f[x_] := {Floor[1 + (n - x[[2]])*x[[1]]], x[[2]] + 1};
a[0] = 0; a[n_] := Nest[f, {1, 0}, n][[1]]/2 (* Joseph E. Cooper III (easonrevant(AT)gmail.com), Aug 19 2008 *) (* updated by Jean-François Alcover, Jun 01 2015 *)
a[n_]:=-(1/2) Subfactorial[-1]-1/2(-1)^n Gamma[2+n] Subfactorial[-2-n]; Table[a[n] //FullSimplify,{n,0,25}] (* Gerry Martens, May 29 2015 *)

A014288(n)=sum(k=0,n,k!)>>1 \\ M. F. Hasler, Dec 16 2007

from math import factorial
def A014288(n): return sum(factorial(k) for k in range(n+1))>>1 # Chai Wah Wu, Nov 01 2023

[sum(factorial(j) for j in (0..n))//2 for n in (0..30)] # G. C. Greubel, Sep 05 2022

a(0)=0, a(1)=1, a(2)=2, a(n) = (n+1)*a(n-1) - n*a(n-2). - Benoit Cloitre, Sep 07 2002
a(0) = 0, a(n) = (1/2)*floor(1 + 1*floor(1 + 2*floor(1 + ... + (n-1)*floor(1+n*floor(1))). - Joseph E. Cooper III (easonrevant(AT)gmail.com), Aug 19 2008
G.f.: G(0)/(1-x)/2 -1/2, where G(k)= 1 + (2*k + 1)*x/( 1 - 2*x*(k+1)/(2*x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
G.f.: A(x) = (Sum_{n>=0} x^n*n!)/(2-2*x) - 1/2 = G(0)/(4*(1-x)) - 1/2, where G(k) = 1 + 1/(1 - x/(x + 1/(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013
a(n) ~ n!/2. - Vaclav Kotesovec, Aug 10 2013
E.g.f.: -1/2 + (exp(x)/2)*Sum_{k>=0} (k! - k*Gamma(k,x)). - Robert Israel, Jun 01 2015
a(n) = ((n+1)!*ExpIntegral(n+2,-1)+Ei(1)+Pi*i)/(2*e). - Ammar Khatab, Aug 14 2020

Edited by M. F. Hasler, Dec 16 2007

A325618 Numbers m such that there exists an integer partition of m whose reciprocal factorial sum is 1.

Original entry on oeis.org

1, 4, 11, 18, 24, 31, 37, 44, 45, 50, 52, 57, 58, 65, 66, 70, 71, 73, 76, 78, 79, 83, 86, 87, 89, 91, 92, 94, 96, 97, 99, 100, 102, 104, 107, 108, 109, 110, 112, 113, 114, 115, 117, 118, 119, 120, 121, 122, 123, 125, 126, 127, 128, 130, 131

Offset: 1

Gus Wiseman, May 13 2019

nonn

The reciprocal factorial sum of an integer partition (y_1,...,y_k) is 1/y_1! + ... + 1/y_k!.

Conjecture: 137 is the greatest integer not in this sequence. - Charlie Neder, May 14 2019

			The sequence of terms together with an integer partition of each whose reciprocal factorial sum is 1 begins:
   1: (1)
   4: (2,2)
  11: (3,3,3,2)
  18: (3,3,3,3,3,3)
  24: (4,4,4,4,3,3,2)
  31: (4,4,4,4,3,3,3,3,3)
  37: (4,4,4,4,4,4,4,4,3,2)
  44: (4,4,4,4,4,4,4,4,3,3,3,3)
  45: (5,5,5,5,5,4,4,4,3,3,2)
  50: (4,4,4,4,4,4,4,4,4,4,4,4,2)

Charlie Neder, Table of n, a(n) for n = 1..1000

Factorial numbers: A000142, A002982, A007489, A011371, A022559, A064986, A115627, A284605, A325616.

Reciprocal factorial sum: A002966, A051908, A058360, A316854, A316855, A325619, A325620, A325621, A325622, A325623, A325624.

a(11)-a(55) from Charlie Neder, May 14 2019

A104344 a(n) = Sum_{k=1..n} k!^2.

Original entry on oeis.org

1, 5, 41, 617, 15017, 533417, 25935017, 1651637417, 133333531817, 13301522971817, 1606652445211817, 231049185247771817, 39006837228880411817, 7639061293780877851817, 1717651314017980301851817, 439480788011413032845851817, 126953027293558583218061851817

Offset: 1

Eric W. Weisstein, Mar 02 2005

Seiichi Manyama, Table of n, a(n) for n = 1..253
Eric Weisstein's World of Mathematics, Factorial Sums

Cf. A001044, A061062, A100289.

Sum_{k=1..n} (k!)^m: A007489 (m=1), this sequence (m=2), A138564 (m=3), A289945 (m=4), A316777 (m=5), A289946 (m=6).

Table[Sum[(k!)^2,{k,n}],{n,15}] (* Harvey P. Dale, Jul 21 2011 *)
Accumulate[(Range[20]!)^2] (* Much more efficient than the above program. *) (* Harvey P. Dale, Aug 15 2022 *)

a(n) = sum(k=1, n, k!^2); \\ Michel Marcus, Jul 16 2017

a(n) = A061062(n) - 1. - Michel Marcus, Feb 28 2014

More terms from Vladimir Joseph Stephan Orlovsky, Sep 24 2009

A166350 Triangle read by rows: T(n,m) = m!, n >= 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 6, 24, 1, 2, 6, 24, 120, 1, 2, 6, 24, 120, 720, 1, 2, 6, 24, 120, 720, 5040, 1, 2, 6, 24, 120, 720, 5040, 40320, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 1, 2, 6, 24, 120, 720, 5040, 40320

Offset: 1

Paul Curtz, Oct 12 2009

			Triangle begins:
  1;
  1, 2;
  1, 2, 6;
  1, 2, 6, 24;
  1, 2, 6, 24, 120;
  1, 2, 6, 24, 120, 720;
  1, 2, 6, 24, 120, 720, 5040;
  ...

Reinhard Zumkeller, Rows n=1..100 of triangle, flattened
Index entries for sequences related to factorial numbers

Cf. A000142, A002260, A094310, A077012, A079210.
Cf. A014454.
Row sums give A007489.

import Data.List (inits)
a166350 n k = a166350_tabl !! (n-1) !! (n-1)
a166350_row n = a166350_tabl !! (n-1)
a166350_tabl = tail $ inits $ tail a000142_list
-- Reinhard Zumkeller, Nov 11 2013

Flatten[Table[Range[n]!,{n,11}]] (* Harvey P. Dale, Jan 06 2012 *)
Module[{nn=20,fs},fs=Range[nn]!;Table[Take[fs,n],{n,nn}]]//Flatten (* Harvey P. Dale, Jun 14 2020 *)

T(n,m) = A000142(m).

Definition clarified - R. J. Mathar, Oct 14 2009

A275736 a(n) has base-2 representation with ones in those digit-positions where n contains ones in its factorial base representation, and zeros in all the other positions.

Original entry on oeis.org

0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 8, 9, 10, 11, 8, 9, 12, 13, 14, 15, 12, 13, 8, 9, 10, 11, 8, 9, 8, 9, 10, 11, 8, 9, 0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 0, 1, 4, 5, 6, 7, 4, 5, 0

Offset: 0

Antti Karttunen, Aug 09 2016

Each natural numbers occurs an infinite number of times.

Can be used when computing A275727.

			22 has factorial base representation "320" (= A007623(22)), which does not contain any "1". Thus a(22) = 0, as the empty sum is 0.
35 has factorial base representation "1121" (= A007623(35)). Here 1's occur in the following positions, when counted from right (starting with 0 for the least significant position): 0, 2 and 3. Thus a(35) = 2^0 + 2^2 + 2^3 = 1*4*8 = 13.

Cf. A000079, A000120, A000225, A007489, A048675, A257261, A257511, A275727, A275730.
Left inverse of A059590.
Cf. A255411 (indices of zeros).
Cf. also A275732.

nn = 120; m = 1; While[Factorial@ m < nn, m++]; m; Map[FromDigits[#, 2] &[IntegerDigits[#, MixedRadix[Reverse@ Range[2, m]]] /. k_ /; k != 1 -> 0] &, Range[0, nn]] (* Michael De Vlieger, Aug 11 2016, Version 10.2 *)

If A257261(n) = 0, then a(n) = 0, otherwise a(n) = A000079(A257261(n)-1) + a(A275730(n, A257261(n)-1)).  [Here A275730(n,p) is a bivariate function that "clears" the digit at zero-based position p in the factorial base representation of n].
Other identities and observations. For all n >= 0:
a(n) = A048675(A275732(n)).
A000120(a(n)) = A257511(n).
a(A007489(n)) = A000225(n).
a(A059590(n)) = n.
a(A255411(n)) = 0.

cp's OEIS Frontend

A180632 Minimum length of a string over the alphabet A = {1,2,...,n} that contains every permutation of A as a substring exactly once, also known as length of the minimal super-permutation.

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A336415 Number of divisors of n! with equal prime multiplicities.

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A227157 Numbers k whose factorial base representation A007623(k) does not contain any nonleading zeros.

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A084556 n occurs n! times.

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A335407 Number of anti-run permutations of the prime indices of n!.

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A014288 a(n) = floor(Sum_{k=0..n} k!/2), or floor( A003422(n+1)/2 ).

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A325618 Numbers m such that there exists an integer partition of m whose reciprocal factorial sum is 1.

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