A038507 -id:A038507 - cp's OEIS frontend

A061057 Factorial splitting: write n! = x*y with x <= y and x maximal; sequence gives value of y-x.

0, 1, 1, 2, 2, 6, 2, 18, 54, 30, 36, 576, 127, 840, 928, 3712, 20160, 93696, 420480, 800640, 1305696, 7983360, 55056804, 65318400, 326592000, 2286926400, 2610934480, 13680979200, 18906930876, 674165366496, 326850970500, 16753029012720, 16880461678080

Offset: 1

Ed Pegg Jr, May 28 2001

nonn

Difference between central divisors of n!. - Jaume Oliver Lafont, Mar 13 2009
For n > 1, n! will never be a square, because of primes in the last half of the factors. Therefore the divisors of n! come in pairs (x,y) with x*y = n! and x < y. The sequence gives the difference y-x between the pair nearest to the square root of n!. - Alois P. Heinz, Jul 06 2009
a(n) = 2 iff n belongs to A146968. - Max Alekseyev, Feb 06 2010

			2! = 1*2, with difference of 1.
3! = 2*3, with difference of 1.
4! = 4*6, with difference of 2.
5! = 10*12, with difference of 2.
6! = 24*30, with difference of 6.
7! = 70*72 with difference of 2.
The corresponding central divisors are two units apart (equivalently, n!+1=A038507(n) is a square) for n = 4, 5, 7 (see A146968).

Max Alekseyev, Table of n, a(n) for n = 1..140

Cf. A000142, A060776, A060777, A060795, A060796, A061060, A061030, A061031, A061032, A061033, A005563, A038507, A038667, A056737, A146968.

A060777 := proc(n) local d,nd ; d := sort(convert(numtheory[divisors](n!),list)) ; nd := nops(d) ; op(floor(1+nd/2),d) ; end:
A060776 := proc(n) local d,nd ; d := sort(convert(numtheory[divisors](n!),list)) ; nd := nops(d) ; op(floor(nd/2),d) ; end:
A061057 := proc(n) A060777(n)-A060776(n) ; end:
seq(A061057(n),n=2..27) ; # R. J. Mathar, Mar 14 2009

Do[ With[ {k = Floor[ Sqrt[ x! ] ] - Do[ If[ Mod[ x!, Floor[ Sqrt[ x! ] ] - n ] == 0, Return[ n ] ], {n, 0, 10000000} ]}, Print[ {x, "! =", k, x!/k, x!/k - k} ] ], {x, 3, 22} ]
f[n_] := Block[{k = Floor@ Sqrt[n! ]}, While[ Mod[n!, k] != 0, k-- ]; n!/k - k]; Table[f@n, {n, 2, 32}] (* Robert G. Wilson v, Jul 11 2009 *)
Table[d=Divisors[n!]; len=Length[d]; If[OddQ[len], 0, d[[1 + len/2]] - d[[len/2]]], {n, 34}] (* Vincenzo Librandi, Jan 02 2016 *)

for(k=2,25,d=divisors(k!);print(d[#d/2+1]-d[#d/2])) \\ Jaume Oliver Lafont, Mar 13 2009

from math import isqrt, factorial
from sympy import divisors
def A061057(n):
    k = factorial(n)
    m = max(d for d in divisors(k,generator=True) if d <= isqrt(k))
    return k//m-m # Chai Wah Wu, Apr 06 2022

a(n) = A060777(n) - A060776(n).

a(n) = A056737(A000142(n)). - Pontus von Brömssen, Jul 15 2023

More terms from Dean Hickerson, Jun 13 2001
Edited by N. J. A. Sloane Jul 07 2009 at the suggestion of R. J. Mathar and Alois P. Heinz
a(41) from Robert G. Wilson v, Oct 03 2014

A089479 Triangle T(n,k) read by rows, where T(n,k) = number of times the permanent of a real n X n (0,1)-matrix takes the value k, for n >= 0, 0 <= k <= n!.

Original entry on oeis.org

0, 1, 1, 1, 9, 6, 1, 265, 150, 69, 18, 9, 0, 1, 27713, 13032, 10800, 4992, 4254, 1440, 1536, 576, 648, 24, 288, 96, 48, 0, 72, 0, 0, 0, 16, 0, 0, 0, 0, 0, 1, 10363361, 3513720, 4339440, 2626800, 3015450, 1451400, 1872800, 962400, 1295700, 425400, 873000

Offset: 0

Hugo Pfoertner, Nov 05 2003

The last element of each row is 1, corresponding to the n X n "all 1" matrix with permanent = n!. The first 4 rows were provided by Wouter Meeussen. The 6th row was computed by Gordon F. Royle: 13906734081, 2722682160, 4513642920, 3177532800, 4466769300, 2396826720, 3710999520, 2065521600, 3253760550, 1468314000, 2641593600, 1350475200, 2210277600, 1034061120,... .

			Triangle begins:
    0,     1;
    1,     1;
    9,     6,     1;
  265,   150,    69,   18,    9,    0,    1;
27713, 13032, 10800, 4992, 4254, 1440, 1536, 576, 648, 24, 288,
                   96, 48, 0, 72, 0, 0, 0, 16, 0, 0, 0, 0, 0, 1;
  ...

Hugo Pfoertner, Table of n, a(n) for n = 0..159 (rows 0..5, flattened)

T(n,0) = A088672(n), T(n,1) = A089482(n). The n-th row of the table contains A087983(n) nonzero entries. For n>2 A089477(n) gives the position of the first zero entry in the n-th row.
Cf. A089480 (occurrence counts for permanents of non-singular (0,1)-matrices), A089481 (occurrence counts for permanents of singular (0,1)-matrices).
Cf. A000290, A038507 (row lengths), A002416 (row sums).
Cf. A036781, A089478.

From Geoffrey Critzer, Dec 20 2023: (Start)
Sum_{k=1..n!} T(n,k) = A227414(n).
For n>2, T(n,n!-(n-1)!) = n^2, the number of matrices with exactly one 0 entry. (End)

Edited by Alois P. Heinz, Dec 20 2023

A002584 Largest prime factor of product of first n primes - 1, or 1 if no such prime exists.

Original entry on oeis.org

1, 5, 29, 19, 2309, 30029, 8369, 929, 46027, 81894851, 876817, 38669, 304250263527209, 92608862041, 59799107, 1143707681, 69664915493, 1146665184811, 17975352936245519, 2140320249725509

Offset: 1

N. J. A. Sloane

The products of the first primes are called primorial numbers. - Franklin T. Adams-Watters, Jun 12 2014

M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors).
M. Kraitchik, Introduction à la Théorie des Nombres. Gauthier-Villars, Paris, 1952, p. 2.
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).

Sean A. Irvine and Amiram Eldar, Table of n, a(n) for n = 1..99 (terms 1..91 from Sean A. Irvine)
A. Borning, Some results for k!+-1 and 2.3.5...p+-1, Math. Comp., 26 (1972), 567-570.
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
S. Kravitz and D. E. Penney, An extension of Trigg's table, Math. Mag., 48 (1975), 92-96.
S. Kravitz and D. E. Penney, An extension of Trigg's table, Math. Mag., 48 (1975), 92-96. [Annotated scanned copy; also letter from N. J. A. Sloane to John Selfridge]
Hisanori Mishima, Factorizations of many number sequences
John Selfridge, Marvin Wunderlich, Robert Morris, N. J. A. Sloane, Correspondence, 1975
R. G. Wilson v, Explicit factorizations.

Cf. A002110, A002585, A006530, A057588.

Prepend[Table[ Max[Transpose[FactorInteger[(Times @@ Prime[Range[i]]) - 1]][[1]]], {i, 2, 20}], 1]
FactorInteger[#][[-1,1]]&/@Rest[FoldList[Times,1,Prime[Range[20]]]-1] (* Harvey P. Dale, Feb 27 2013 *)

a(n)=if(n>1, my(f=factor(prod(i=1,f,prime(i)))[,1]); f[#f], 1) \\ Charles R Greathouse IV, Feb 08 2017

a(n) = A006530(A057588(n)). - Amiram Eldar, Feb 13 2020

More terms from J. L. Selfridge

Further terms from Labos Elemer, Oct 25 2000

A052898 a(n) = 2*n! + 1.

Original entry on oeis.org

3, 3, 5, 13, 49, 241, 1441, 10081, 80641, 725761, 7257601, 79833601, 958003201, 12454041601, 174356582401, 2615348736001, 41845579776001, 711374856192001, 12804747411456001, 243290200817664001

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 874

Cf. sequences of the type k*n!+1: A038507 (k=1), this sequence, A173324 (k=3), A173322 (k=4), A173319 (k=5), A173314 (k=6).

[2*Factorial(n) + 1: n in [0..20]]; // Vincenzo Librandi, Sep 29 2013

[3] cat [n eq 1 select n+2 else n*Self(n-1)-n+1: n in [1..25] ]; // Vincenzo Librandi, Sep 29 2013

spec := [S,{S=Union(Sequence(Z),Sequence(Z),Set(Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
a[0]:=3: for n from 1 to 21 do a[n]:=n*a[n-1]-n+1; od:
seq(a[n], n=0..20). # Sergei N. Gladkovskii, Jul 04 2012

lst={};s=-3;Do[s+=(n+=s*n);AppendTo[lst, s], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 10 2008 *)
FoldList[#1*#2 - #2 + 1 &, 3, Range[19]] (* Robert G. Wilson v, Jul 07 2012 *)
Table[2 n! + 1, {n, 0, 20}] (* Vincenzo Librandi, Sep 29 2013 *)

E.g.f.: (-2-exp(x)+x*exp(x))/(-1+x).
Recurrence: {a(2)=5, a(1)=3, (n^2+2*n+1)*a(n)+(-n^2-3*n-1)*a(n+1)+a(n+2)*n}
From Sergei N. Gladkovskii, Jul 04 2012: (Start)
a(0)=3; for n>0, a(n) = n*a(n-1)-n+1.
Let E(x) be the e.g.f., then
E(x)=(x*G(0)-2)/(x-1), where G(k)= 1 - 1/(x - x^3/(x^2 - (k+1)/G(k+1)));(continued fraction, 3rd kind, 3-step).
E(x)=x*G(0)/(x-1), where G(k)= 1 - 1/(x + 2*x*(x-1)*k!/(1 - 2*(x-1)*k! - x^2/(x^2 + 2*(x-1)*(k+1)!/G(k+1)))); (continued fraction, 3rd kind, 4-step).
(End).

Definition replaced with the closed formula by Bruno Berselli, Sep 28 2013

A057713 Smallest prime divisor of Kummer numbers ( = primorials - 1), or 1 if no such prime exists.

Original entry on oeis.org

1, 5, 29, 11, 2309, 30029, 61, 53, 37, 79, 228737, 229, 304250263527209, 141269, 191, 87337, 27600124633, 1193, 163, 260681003321, 313, 163, 139, 23768741896345550770650537601358309, 66683, 2990092035859, 15649, 17515703, 719, 295201, 15098753, 10172884549, 20962699238647, 4871, 673, 311, 1409, 1291, 331, 1450184819, 23497, 711427, 521, 673, 519577, 1372062943, 56543, 811, 182309, 53077, 641, 349, 389

Offset: 1

Labos Elemer, Oct 25 2000

nonn

			6th term in the sequence corresponds to 7th primorial = 510510 and 510509 = 61 * 8369, so a(7) = 61.

Amiram Eldar, Table of n, a(n) for n = 1..99
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018. - _N. J. A. Sloane_, Jun 13 2012
Hisanori Mishima, Factorizations of many number sequences
R. G. Wilson v, Explicit factorizations

Cf. A002110, A002584, A002585, A006862, A020639, A057588.

Map[If[PrimeQ@ #, #, FactorInteger[#][[1, 1]]] &, FoldList[#1 #2 &, Prime@ Range@ 36] - 1] (* Michael De Vlieger, Feb 18 2017 *)

a(n) = A020639(A057588(n)). - Amiram Eldar, Feb 13 2020

More terms from Klaus Brockhaus, Larry Reeves (larryr(AT)acm.org) and Robert G. Wilson v, Apr 02 2001

A173314 a(n) = 6*n!+1.

Original entry on oeis.org

7, 7, 13, 37, 145, 721, 4321, 30241, 241921, 2177281, 21772801, 239500801, 2874009601, 37362124801, 523069747201, 7846046208001, 125536739328001, 2134124568576001, 38414242234368001, 729870602452992001

Offset: 0

Vincenzo Librandi, Feb 16 2010

nonn

			For n=0, a(0)=7; n=1,a(1)=7; n=2, a(2)=13; n=3, a(3)=37; n=4, a(4)=145.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. sequences of the type k*n!+1: A038507 (k=1), A052898 (k=2), A173324 (k=3), A173322 (k=4), A173319 (k=5), this sequence (k=6).

[6*Factorial(n) + 1: n in [0..25]]; // Vincenzo Librandi, Sep 29 2013

[7] cat [n eq 1 select n+6 else n*Self(n-1)-n+1: n in [1..25] ]; // Vincenzo Librandi, Sep 29 2013

Table[6 n! + 1, {n, 0, 25}] (* Vincenzo Librandi, Sep 29 2013 *)

a(0)=7, a(n) = n*a(n-1)-n+1. - Vincenzo Librandi, Sep 29 2013

A173322 a(n) = 4*n! + 1.

Original entry on oeis.org

5, 5, 9, 25, 97, 481, 2881, 20161, 161281, 1451521, 14515201, 159667201, 1916006401, 24908083201, 348713164801, 5230697472001, 83691159552001, 1422749712384001, 25609494822912001, 486580401635328001, 9731608032706560001, 204363768686837760001

Offset: 0

Vincenzo Librandi, Feb 16 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. sequences of the type k*n!+1: A038507 (k=1), A052898 (k=2), A173324 (k=3), this sequence, A173319 (k=5), A173314 (k=6).

[4*Factorial(n) + 1: n in [0..25]]; // Vincenzo Librandi, Sep 29 2013

[5] cat [n eq 1 select n+4 else n*Self(n-1)-n+1: n in [1..25] ]; // Vincenzo Librandi, Sep 29 2013

a:= proc(n) if n=0 then 5 else a(n) := n*a(n-1)-n+1 fi end: seq (a(n), n=0..25);  # Sergei N. Gladkovskii, Jul 04 2012

4*Range[0,20]!+1 (* Harvey P. Dale, Jun 26 2012 *)
Table[4 n! + 1, {n, 0, 21}] (* Vincenzo Librandi, Sep 29 2013 *)

a(0) = 5, a(n) = n*a(n-1)-n+1. - Sergei N. Gladkovskii, Jul 04 2012

A173324 a(n) = 3*n! + 1.

Original entry on oeis.org

4, 4, 7, 19, 73, 361, 2161, 15121, 120961, 1088641, 10886401, 119750401, 1437004801, 18681062401, 261534873601, 3923023104001, 62768369664001, 1067062284288001, 19207121117184001, 364935301226496001, 7298706024529920001, 153272826515128320001

Offset: 0

Vincenzo Librandi, Feb 16 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. sequences of the type k*n!+1: A038507 (k=1), A052898 (k=2), this sequence, A173322 (k=4), A173319 (k=5), A173314 (k=6).

[3*Factorial(n) + 1: n in [0..25]]; // Vincenzo Librandi, Sep 29 2013

[4] cat [n eq 1 select n+3 else n*Self(n-1)-n+1: n in [1..25] ]; // Vincenzo Librandi, Sep 29 2013

a:= proc(n) if n=0 then 4 else a(n):= n*a(n-1)-n+1 fi end: seq (a(n), n=0..25);  # Sergei N. Gladkovskii, Jul 04 2012

Table[3 n! + 1, {n, 0, 30}] (* Vincenzo Librandi, Sep 29 2013 *)

a(0) = 4, a(n) = n*a(n-1)-n+1 for n>0. - Sergei N. Gladkovskii, Jul 04 2012

A229345 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component or all components by the same positive integer; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 7, 22, 4, 1, 1, 25, 248, 188, 8, 1, 1, 121, 6506, 11380, 1712, 16, 1, 1, 721, 292442, 2359348, 577124, 16098, 32, 1, 1, 5041, 19450082, 1088626684, 991365512, 30970588, 154352, 64, 1

Offset: 0

Alois P. Heinz, Sep 24 2013

			A(2,2) = 22: [(2,2),(1,1),(0,0)], [(2,2),(1,1),(0,1),(0,0)], [(2,2),(1,1),(1,0),(0,0)], [(2,2),(0,0)], [(2,2),(1,2),(0,1),(0,0)], [(2,2),(1,2),(0,2),(0,1),(0,0)], [(2,2),(1,2),(0,2),(0,0)], [(2,2),(1,2),(1,1),(0,0)], [(2,2),(1,2),(1,1),(0,1),(0,0)], [(2,2),(1,2),(1,1),(1,0),(0,0)], [(2,2),(1,2),(1,0),(0,0)], [(2,2),(0,2),(0,1),(0,0)], [(2,2),(0,2),(0,0)], [(2,2),(2,1),(1,0),(0,0)], [(2,2),(2,1),(1,1),(0,0)], [(2,2),(2,1),(1,1),(0,1),(0,0)], [(2,2),(2,1),(1,1),(1,0),(0,0)], [(2,2),(2,1),(0,1),(0,0)], [(2,2),(2,1),(2,0),(1,0),(0,0)], [(2,2),(2,1),(2,0),(0,0)], [(2,2),(2,0),(1,0),(0,0)], [(2,2),(2,0),(0,0)].
Square array A(n,k) begins:
  1,  1,     1,        1,            1,                 1, ...
  1,  1,     3,        7,           25,               121, ...
  1,  2,    22,      248,         6506,            292442, ...
  1,  4,   188,    11380,      2359348,        1088626684, ...
  1,  8,  1712,   577124,    991365512,     4943064622568, ...
  1, 16, 16098, 30970588, 453530591824, 25162900228200976, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened

Columns k=0-3 give: A000012, A011782, A132595(n+1), A229482.
Rows n=0-2 give: A000012, A038507 (for k>1), A229465.
Main diagonal gives: A229346.
Cf. A060854, A227578, A227655, A225094, A210472, A229142, A262809, A263159.

b:= proc(l) option remember; local m; m:= nops(l);
      `if`(m=0 or l[m]=0, 1,
      `if`(m>1, add(b(l-[j$m]), j=1..l[1]), 0)+
      add(add(b(sort(subsop(i=l[i]-j, l))), j=1..l[i]), i=1..m))
    end:
A:= (n, k)-> b([n$k]):
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Sep 24 2013

b[l_] := b[l] = With[{m = Length[l]}, If[m == 0 || l[[m]] == 0, 1, If[m > 1, Sum[b[l - Array[j&, m]], {j, 1, l[[1]]}],  0] + Sum[Sum[b[Sort[ReplacePart[l, i -> l[[i]] - j]]], {j, 1, l[[i]]}], {i, 1, m}]]]; a[n_, k_] := b[Array[n&, k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)

A054415 Smallest prime factor of n!-1 (for n>2), a(2)=1.

Original entry on oeis.org

1, 5, 23, 7, 719, 5039, 23, 11, 29, 13, 479001599, 1733, 87178291199, 17, 3041, 19, 59, 653, 124769, 23, 109, 51871, 625793187653, 149, 20431, 29, 239, 31, 265252859812191058636308479999999, 787, 263130836933693530167218012159999999, 8683317618811886495518194401279999999

Offset: 2

Henry Bottomley, May 10 2000

nonn

The initial term a(2)=1 is not a prime, but it does not affect search results and may prevent submission of duplicates. - M. F. Hasler, Oct 31 2012

			a(3)=5 because 3!-1=5 which is prime; a(5)=7 because 5!-1=119=7*17 and 7<17

Chai Wah Wu, Table of n, a(n) for n = 2..153 (n = 2..135 from Amiram Eldar)
P. Erdős and C. L. Stewart, On the greatest and least prime factors of n! + 1, J. London Math. Soc. (2) 13:3 (1976), pp. 513-519.
M. Kraitchik, On the divisibility of factorials, Scripta Math., 14 (1948), 24-26 (but beware errors). [Annotated scanned copy]
Hisanori Mishima, Factorizations of many number sequences: n! - 1 (n = 1 to 100); Primorials - 1.
R. G. Wilson v, Explicit factorizations

Cf. A002582, A033312, A051301.

Do[ Print[ FactorInteger[ n! - 1, FactorComplete -> True][ [1, 1] ] ], {n, 3, 32} ]

A054415(n)=if(n>2,factor(n!-1)[1,1],1)  \\ M. F. Hasler, Oct 31 2012

Erdős & Stewart show that a(n) > n + (l-o(l))log n/log log n except when n+1 is prime, and that a(n) > n + e(n)sqrt(n) for almost all n where e(n) is any function with lim e(n) = 0. - Charles R Greathouse IV, Dec 05 2012

More terms from Robert G. Wilson v, Aug 01 2000

More terms from Amiram Eldar, Oct 07 2019

cp's OEIS Frontend

A061057 Factorial splitting: write n! = x*y with x <= y and x maximal; sequence gives value of y-x.

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A089479 Triangle T(n,k) read by rows, where T(n,k) = number of times the permanent of a real n X n (0,1)-matrix takes the value k, for n >= 0, 0 <= k <= n!.

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A002584 Largest prime factor of product of first n primes - 1, or 1 if no such prime exists.

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A052898 a(n) = 2*n! + 1.

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A057713 Smallest prime divisor of Kummer numbers ( = primorials - 1), or 1 if no such prime exists.

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A173314 a(n) = 6*n!+1.

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A173322 a(n) = 4*n! + 1.

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