A061030 -id:A061030 - cp's OEIS frontend

A060796 Upper central divisor of n-th primorial.

2, 3, 6, 15, 55, 182, 715, 3135, 15015, 81345, 448630, 2733549, 17490603, 114388729, 785147363, 5708795638, 43850489690, 342503171205, 2803419704514, 23622001517543, 201817933409378, 1793779635410490, 16342166369958702, 154171363634898185, 1518410187442699518, 15259831781575946565

Offset: 1

Labos Elemer, Apr 27 2001

nonn

Also: Write product of first n primes as x*y with x < y and x maximal; sequence gives value of y. Indeed, p(n)# = primorial(n) = A002110(n) is never a square for n >= 1; all exponents in the prime factorization are 1. Therefore primorial(n) has N = 2^n distinct divisors. Since this is an even number, the N divisors can be grouped in N/2 pairs {d(k), d(N+1-k)} with product equal to p(n)#. One of the two is always smaller and one is larger than sqrt(p(n)#). This sequence gives the (2^(n-1)+1)-th divisor, which is the smallest one larger than sqrt(p(n)#). - M. F. Hasler, Sep 20 2011

			n = 8, q(8) = 2*3*5*7*11*13*17*19 = 9699690. Its 128th and 129th divisors are {3094, 3135}: a(8) = 3135, and 3094 < A000196(9699690) = 3114 < 3135. [Corrected by _M. F. Hasler_, Sep 20 2011]

Max Alekseyev, Table of n, a(n) for n = 1..70

Cf. A060755, A000196, A002110, A033677, A060776, A060777, A061057, A060795 (x), A061060 (y-x), A182987 (x+y), A061030, A061031, A061032, A061033, A200744.

k = 1; Do[k *= Prime[n]; l = Divisors[k]; x = Length[l]; Print[l[[x/2 + 1]]], {n, 1, 24}] (* Ryan Propper, Jul 25 2005 *)

A060796(n) = divisors(prod(k=1,n,prime(k)))[2^(n-1)+1] \\ Requires stack size > 2^(n+5). - M. F. Hasler, Sep 20 2011

a(n) = A033677(A002110(n)).

a(n) = A002110(n)/A060795(n). - M. F. Hasler, Mar 21 2022

More terms from Ryan Propper, Jul 25 2005
a(24)-a(37) in b-file calculated from A182987 by M. F. Hasler, Sep 20 2011
a(38) from David A. Corneth, Mar 21 2022
a(39)-a(70) in b-file from Max Alekseyev, Apr 20 2022

A060795 Write product of first n primes as x*y with x

Original entry on oeis.org

1, 2, 5, 14, 42, 165, 714, 3094, 14858, 79534, 447051, 2714690, 17395070, 114371070, 783152070, 5708587335, 43848093003, 342444658094, 2803119896185, 23619540863730, 201813981102615, 1793779293633437, 16342050964565645, 154170926013430326, 1518409177581024365

Offset: 1

Labos Elemer, Apr 27 2001

nonn

Or, lower central divisor of n-th primorial.

Subsequence of A005117 (squarefree numbers). - Michel Marcus, Feb 22 2016

			n = 8: q(8) = 2*3*5*7*11*13*17*19 = 9699690. Its 128th and 129th divisors are {3094, 3135}: a(8) = 3094 and 3094 < A000196(9699690) = 3114 < 3135. [Corrected by _Colin Barker_, Oct 22 2010]
2*3*5*7 = 210 = 14*15 with difference of 1, so a(4) = 14.

Max Alekseyev, Table of n, a(n) for n = 1..70

Cf. A000196, A060776, A060777, A061057, A060796 (y), A061060 (y-x), A182987 (x+y), A061030, A061031, A061032, A061033, A060755, A033677, A200743.

F:= proc(n) local P,N,M;
     P:= {seq(ithprime(i),i=1..n)};
     N:= floor(sqrt(convert(P,`*`)));
     M:= map(convert, combinat:-powerset(P),`*`);
     max(select(`<=`,M,N))
end proc:
map(F, [$1..20]); # Robert Israel, Feb 22 2016

a[n_] := (m = Times @@ Prime[Range[n]] ; dd = Divisors[m]; dd[[Length[dd]/2 // Floor]]); Table[Print[an = a[n]]; an, {n, 1, 25}] (* Jean-François Alcover, Oct 15 2016 *)

a(n) = my(m=prod(i=1, n, prime(i))); divisors(m)[numdiv(m)\2]; \\ Michel Marcus, Feb 22 2016

a(n) = A060775(A002110(n)). - Labos Elemer, Apr 27 2001

a(n) = A002110(n)/A060796(n). - M. F. Hasler, Mar 21 2022

More terms from Ed Pegg Jr, May 28 2001
a(16)-a(23) computed by Jud McCranie, Apr 15 2000
a(24) and a(25) from Robert Israel, Feb 22 2016
a(25) corrected by Jean-François Alcover, Oct 15 2016
a(26)-a(33) in b-file from Amiram Eldar, Apr 09 2020
Up to a(38) using b-file of A060796, by M. F. Hasler, Mar 21 2022
a(39)-a(70) in b-file from Max Alekseyev, Apr 20 2022

A061033 Factorial splitting: write n! = xyz with x

Original entry on oeis.org

2, 2, 2, 2, 6, 4, 17, 24, 30, 42, 72, 112, 288, 420, 720, 1568, 1920, 3512, 16560, 19686, 16028, 71280, 182160, 184320, 552960, 2925648, 4885160, 12241152, 40191471, 71559680, 77631750, 217165824, 604653336, 368858880

Offset: 3

Ed Pegg Jr, May 25 2001

nonn

Luc Kumps, personal communication.

Max Alekseyev, Table of n, a(n) for n = 3..100

Cf. A061030 - A061032.

a(10) and a(11) corrected and a(14)-a(31) from Donovan Johnson, May 11 2010

Definition and a(14), a(18), a(24) are corrected, terms a(32) onward added by Max Alekseyev, Apr 10 2022

A061032 Factorial splitting: write n! = xyz with x

Original entry on oeis.org

3, 4, 6, 10, 21, 36, 81, 168, 360, 810, 1872, 4480, 11088, 27720, 71280, 186368, 496128, 1347192, 3720960, 10407936, 29576988, 85322160, 249500160, 738904320, 2216712960, 6732000000, 20680540160, 64260000000, 201860859375

Offset: 3

Ed Pegg Jr, May 25 2001

nonn

We first maximize x and then minimize z, which may be different from doing the opposite way around. For example, 7! = 15*16*21 = 14*18*20 is the case when absolute maximum of x (=15) and absolute minimum of z (=20) cannot be achieved together. - Max Alekseyev, Jun 18 2022

Luc Kumps, personal communication.

Max Alekseyev, Table of n, a(n) for n = 3..100

Cf. A061030, A061031, A061033, A060776, A060777, A060795, A060796, A200743, A200744.

a(10) and a(11) corrected and a(14)-a(31) from Donovan Johnson, May 11 2010

Definition and a(14), a(18), a(24) are corrected by Max Alekseyev, Apr 10 2022

A061031 Factorial splitting: write n! = xyz with x

Original entry on oeis.org

2, 3, 5, 9, 16, 35, 70, 150, 336, 770, 1848, 4455, 10920, 27648, 70720, 185895, 496125, 1344000, 3706560, 10395840, 29568000, 85299200, 249356800, 738840960, 2216522880, 6730407936, 20678434920, 64248260076, 201838500864

Offset: 3

Ed Pegg Jr, May 25 2001

nonn

We first maximize x and then minimize z, which may be different from doing the opposite way around. For example, 7! = 15*16*21 = 14*18*20 is the case when absolute maximum of x (=15) and absolute minimum of z (=20) cannot be achieved together. - Max Alekseyev, Jun 18 2022

Luc Kumps, personal communication.

Max Alekseyev, Table of n, a(n) for n = 3..100

Cf. A061030, A061032, A061033, A060776, A060777, A060795, A060796, A200743, A200744.

a(10) and a(11) corrected and a(14)-a(31) from Donovan Johnson, May 11 2010

Definition and a(14), a(18), a(24) are corrected by Max Alekseyev, Apr 10 2022

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

Original entry on oeis.org

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

A061060 Write product of first n primes as x*y with x

Original entry on oeis.org

1, 1, 1, 1, 13, 17, 1, 41, 157, 1811, 1579, 18859, 95533, 17659, 1995293, 208303, 2396687, 58513111, 299808329, 2460653813, 3952306763, 341777053, 115405393057, 437621467859, 1009861675153, 6660853109087, 29075165225531

Offset: 1

Ed Pegg Jr, May 28 2001

nonn

			a(4)=1: 2*3*5*7 = 210 = 14*15, so we can take x=14, y=15, with difference of 1.
Also: n=3: 2*3-5=1; n=4: 3*5-2*7=1; n=5: 5*11-2*3*7=13; n=6: 2*7*13-3*5*11=17; n=7: 5*11*13-2*3*7*17=1; n=8: 3*5*11*19-2*7*13*17=41

Max Alekseyev, Table of n, a(n) for n = 1..70
C. Aebi and G. Cairns, Partitions of primes, Parabola 45, Issue 1 (2009). - _Jonathan Sondow_, Jun 21 2012
Carlos Rivera, Conjecture 18. Minimal Primorial Partitions, The Prime Puzzles & Problems Connection.

Cf. A038667, A060776, A060777, A061057, A060795, A060796, A061030-A061033, A182987, A263292, A352813.

A061060aux := proc(l1,l2) local resul ; resul := product(l1[i],i=1..nops(l1)) ; resul := resul-product(l2[i],i=1..nops(l2)) ; RETURN(abs(resul)) ; end:
A061060 := proc(n) local plist,i,subl,resul,j,l1,l2,k,d ; plist := [] ; resul := 1 ; for i from 1 to n do resul := resul*ithprime(i) ; plist := [op(plist), ithprime(i)] ; od; for i from 1 to n/2 do subl := combinat[choose](plist,i) ; for j from 1 to nops(subl) do l1 := op(j,subl) ; l2 := convert(plist,set) minus convert(l1,set) ; d := A061060aux(l1,l2) ; if d < resul then resul := d ; fi ; od; od ; RETURN(resul) ; end:
for n from 3 to 19 do printf("%d,",A061060(n)) ; od ; # R. J. Mathar, Aug 26 2006 [This Maple program was attached to A121315. However I think it belongs here, so I renamed the variables and moved it to this entry. - N. J. A. Sloane, Sep 16 2005]

(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) f[n_] := Block[{arrayofnprimes = Array[Prime, n], primorial = Times @@ Array[Prime, n], diffmin = Infinity, adiff, sub}, If[n == 1, 1, Do[sub = Times @@ NthSubset[i, arrayofnprimes]; adiff = Abs[primorial/sub - sub]; If[adiff < diffmin, diffmin = adiff], {i, 2, 2^n/2}]; diffmin]]; Do[ Print@f@n, {n, 30}] (* Robert G. Wilson v, Sep 14 2006 *)

Conjecture: Limit_{N->oo} (Sum_{n=1..N} log(a(n))) / (Sum_{n=1..N} prime(n)) = 1/e (A068985). - Alain Rocchelli, Nov 13 2023

Terms a(16)-a(45) in b-file computed by Jud McCranie, Apr 15 2000; Jan 12 2016
a(46)-a(60) in b-file from Don Reble, Jul 11 2020
a(61)-a(70) in b-file from Max Alekseyev, Apr 20 2022

A355189 Factorial splitting: write n! = xyz with x <= y <= z and minimal z-x; sequence gives value of x.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 8, 14, 32, 70, 140, 324, 768, 1800, 4368, 10800, 27300, 70560, 184800, 494208, 1343680, 3704400, 10388250, 29560960, 85250880, 249318000, 738720000, 2216160000, 6729074352, 20675655000, 64245312000, 201819656500, 640760440320

Offset: 0

Max Alekseyev, Jun 23 2022

nonn

Apparently we have x < y < z for all n > 9. If so, using strict inequalities x < y < z in the definition would make the sequence undefined for n < 3 and affect only a(9) by switching from 9! = 70*72*72 to 9! = 63*72*80.

Max Alekseyev, Table of n, a(n) for n = 0..100

Cf. A061030, A061031, A061032, A061033, A060776, A060777, A060795, A060796, A200743, A200744, A355190, A355191, A355192.

A355190 Factorial splitting: write n! = xyz with x <= y <= z and minimal z-x; sequence gives value of y.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 18, 35, 72, 160, 350, 770, 1848, 4455, 10920, 27648, 70720, 185895, 496125, 1344000, 3706560, 10395840, 29568000, 85299200, 249356800, 738840960, 2216522880, 6730407936, 20678434920, 64253314125, 201847852800, 640813814784, 2055410286592, 6658705461408, 21780889600000

Offset: 0

Max Alekseyev, Jun 23 2022

nonn

Max Alekseyev, Table of n, a(n) for n = 0..100

Cf. A061030, A061031, A061032, A061033, A060776, A060777, A060795, A060796, A200743, A200744, A355189, A355191, A355192.

A355191 Factorial splitting: write n! = xyz with x <= y <= z and minimal z-x; sequence gives value of z.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 10, 20, 36, 72, 162, 352, 810, 1872, 4480, 11088, 27720, 71280, 186368, 496128, 1347192, 3720960, 10407936, 29576988, 85322160, 249500160, 738904320, 2216712960, 6732000000, 20680540160, 64257392640, 201852518400, 640832000000, 2055425699250, 6658777165824, 21781337550336

Offset: 0

Max Alekseyev, Jun 23 2022

nonn

Apparently we have x < y < z for all n > 9. If so, using strict inequalities x < y < z in the definition would make the sequence undefined for n < 3 and affect only a(9) by switching from 9! = 70*72*72 to 9! = 63*72*80.

Max Alekseyev, Table of n, a(n) for n = 0..100

Cf. A061030, A061031, A061032, A061033, A060776, A060777, A060795, A060796, A200743, A200744, A355189, A355190, A355192.

cp's OEIS Frontend

A060796 Upper central divisor of n-th primorial.

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A060795 Write product of first n primes as x*y with x

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A061033 Factorial splitting: write n! = x*y*z with x

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A061032 Factorial splitting: write n! = x*y*z with x

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A061031 Factorial splitting: write n! = x*y*z with x

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

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A061060 Write product of first n primes as x*y with x

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A061033 Factorial splitting: write n! = xyz with x

A061032 Factorial splitting: write n! = xyz with x

A061031 Factorial splitting: write n! = xyz with x

A355189 Factorial splitting: write n! = xyz with x <= y <= z and minimal z-x; sequence gives value of x.

A355190 Factorial splitting: write n! = xyz with x <= y <= z and minimal z-x; sequence gives value of y.

A355191 Factorial splitting: write n! = xyz with x <= y <= z and minimal z-x; sequence gives value of z.