A200743 -id:A200743 - cp's OEIS frontend

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

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

A061030 Factorial splitting: write n! = xyz with x

Original entry on oeis.org

1, 2, 4, 8, 15, 32, 64, 144, 330, 768, 1800, 4368, 10800, 27300, 70560, 184800, 494208, 1343680, 3704400, 10388250, 29560960, 85250880, 249318000, 738720000, 2216160000, 6729074352, 20675655000, 64247758848, 201820667904

Offset: 3

Ed Pegg Jr, May 25 2001

nonn

			For n = 6, 6! = 720 = 8*9*10, so x=8, y=9, z=10.

Luc Kumps, personal communication.

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

Cf. A061031, 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

A200744 Divide integers 1..n into two sets, minimizing the difference of their products. This sequence is the larger product.

Original entry on oeis.org

1, 2, 3, 6, 12, 30, 72, 210, 630, 1920, 6336, 22176, 79200, 295680, 1146600, 4586400, 18869760, 80061696, 348986880, 1560176640, 7148445696, 33530112000, 160825785120, 787718131200, 3938590656000, 20083261440000, 104351247000000, 552173794099200, 2973528918360000, 16286983961149440

Offset: 1

Franklin T. Adams-Watters, Nov 21 2011

nonn

			For n=1, we put 1 in one set and the other is empty; with the standard convention for empty products, both products are 1.
For n=13, the central pair of divisors of n! are 78975 and 78848. Since neither is divisible by 10, these values cannot be obtained. The next pair of divisors are 79200 = 12*11*10*6*5*2*1 and 78624 = 13*9*8*7*4*3, so a(13) = 79200.

Max Alekseyev, Table of n, a(n) for n = 1..140 (terms for n = 1..35 from Michael S. Branicky)

Cf. A060777, A060796, A038667, A127180, A200743.

a:= proc(n) local l, ll, g, p, i; l:= [i$i=1..n]; ll:= [i!$i=1..n]; g:= proc(m, j, b) local mm, bb, k; if j=1 then m else mm:= m; bb:= b; for k to 2 while (mmbb then bb:= max(bb, g(mm, j-1, bb)) fi; mm:= mm*l[j] od; bb fi end; Digits:= 700; p:= ceil(sqrt(ll[n])); ll[n]/ g(1, nops(l), 1) end: seq(a(n), n=1..23);  # Alois P. Heinz, Nov 22 2011

a[n_] := a[n] = Module[{s, t}, {s, t} = MinimalBy[{#, Complement[Range[n], #]}& /@ Subsets[Range[n]], Abs[Times @@ #[[1]] - Times @@ #[[2]]]&][[1]]; Max[Times @@ s, Times @@ t]];
Table[Print[n, " ", a[n]];
a[n], {n, 1, 25}] (* Jean-François Alcover, Nov 07 2020 *)

from math import prod, factorial
from itertools import combinations
def A200744(n):
    m = factorial(n)
    return min((abs((p:=prod(d))-m//p),max(p,m//p)) for l in range(n,n//2,-1) for d in combinations(range(1,n+1),l))[1] # Chai Wah Wu, Apr 07 2022

a(n) = A127180(n) - A200743(n) = A038667(n) + A200743(n) = (A038667(n) + A127180(n)) / 2. - Max Alekseyev, Jun 18 2022

a(24)-a(30) from Alois P. Heinz, Nov 22 2011

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

A038667 Minimal value of |product(A) - product(B)| where A and B are a partition of {1,2,3,...,n}.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 6, 2, 18, 54, 30, 36, 576, 576, 840, 6120, 24480, 20160, 93696, 420480, 800640, 1305696, 7983360, 80313120, 65318400, 326592000, 2286926400, 3002360256, 13680979200, 37744574400, 797369149440, 1763653953600, 16753029012720, 16880461678080, 10176199188480, 26657309952000

Offset: 0

Erich Friedman

nonn

Conjecture: The sequence of rational numbers A061057(n)/a(n) has 1 as a limit point. Question: What other limit points does the sequence have? - Richard Peterson, Jul 13 2023

			For n=1, we put 1 in one set and the other is empty; with the standard convention for empty products, both products are 1.
For n=13, the central pair of divisors of n! are 78975 and 78848. Since neither is divisible by 10, these values cannot be obtained. The next pair of divisors are 79200 = 12*11*10*6*5*2*1 and 78624 = 13*9*8*7*4*3, so a(13) = 79200 - 78624 = 576.

Max Alekseyev, Table of n, a(n) for n = 0..140 (terms for n = 0..60 from Peter J. Taylor)

Cf. A061033, A061057, A061060, A200743, A200744, A127180, A263292.

a:= proc(n) local l, ll, g, gg, p, i; l:= [i$i=1..n]; ll:= [i!$i=1..n]; g:= proc(m, j, b) local mm, bb, k; if j=1 then m else mm:= m; bb:= b; for k to 2 while (mmbb then bb:= max(bb, g(mm, j-1, bb)) fi; mm:= mm*l[j] od; bb fi end; Digits:= 700; p:= ceil(sqrt(ll[n])); gg:= g(1, nops(l), 1); ll[n]/gg -gg end: a(0):=0:
seq(a(n), n=0..20); #  Alois P. Heinz, Jul 09 2009, revised Oct 17 2015

a[n_] := Module[{l, ll, g, gg, p, i}, l = Range[n]; ll = Array[Factorial, n]; g[m_, j_, b_] := g[m, j, b] = Module[{mm, bb, k}, If[j==1, m, mm=m; bb=b; For[k=1, mm bb , bb = Max[bb, g[mm, j-1, bb]]]; mm = mm*l[[j]]]; bb]]; p = Ceiling[Sqrt[ ll[[n]]]]; gg = g[1, Length[l], 1]; ll[[n]]/gg - gg]; a[0]=0; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 35}] (* Jean-François Alcover, Feb 29 2016, after Alois P. Heinz *)

from math import prod, factorial
from itertools import combinations
def A038667(n):
    m = factorial(n)
    return 0 if n == 0 else min(abs((p:=prod(d))-m//p) for l in range(n,n//2,-1) for d in combinations(range(1,n+1),l)) # Chai Wah Wu, Apr 06 2022

a(n) = A200744(n) - A200743(n) = (A200744(n)^2 - A200743(n)^2) / A127180(n). - Max Alekseyev, Apr 08 2022

a(n) >= A061057(n).

a(28)-a(31) from Alois P. Heinz, Jul 09 2009
a(1) and examples from Franklin T. Adams-Watters, Nov 22 2011
a(32)-a(33) from Alois P. Heinz, Nov 23 2011
a(34)-a(35) from Alois P. Heinz, Oct 17 2015

A127180 a(n) = smallest possible (product of b(k)'s + product of c(k)'s), where the positive integers <= n are partitioned somehow into {b(k)} and {c(k)}.

Original entry on oeis.org

2, 2, 3, 5, 10, 22, 54, 142, 402, 1206, 3810, 12636, 43776, 157824, 590520, 2287080, 9148320, 37719360, 160029696, 697553280, 3119552640, 14295585696, 67052240640, 321571257120, 1575370944000, 7876854720000, 40164235953600

Offset: 0

Leroy Quet, Jan 07 2007

nonn

The maximum (product of b(k)'s + product of c(k)'s) occurs, for n>=2, when {b(k)} = (2,3,4,...n) and {c(k)} = (1). a(1) = 2 because the product over the empty set is defined here as 1.

			By partitioning (1,2,3,...8) into {b(k)} and {c(k)} so that {b(k)} = (1,4,6,8) and {c(k)} = (2,3,5,7), then (product of b(k)'s + product of c(k)'s) is minimized. Therefore a(8) = 1*4*6*8 + 2*3*5*7 = 402.

Max Alekseyev, Table of n, a(n) for n = 0..140

Cf. A038667, A127181, A200743, A200744,

LQprod := proc(S) if nops(S) = 0 then 1 ; else product(S[i],i=1..nops(S)) ; fi ; end: A127180 := proc(n) local S,m,B,b,c,s,res,i ; res := -1 ; S := {} ; for i from 1 to n do S := S union {i} ; od; for m from 0 to n/2 do B := combinat[permute](n,m) ; for i from 1 to nops(B) do b := op(i,B) ; c := S minus convert(b,set) ; s := LQprod(b)+LQprod(c) ; if res < 0 or s < res then res := s ; fi ; od ; od ; RETURN(res) ; end: for n from 1 to 20 do A127180(n) ; od ; # R. J. Mathar, Jan 10 2007

a[n_] := a[n] = Module[{s, t}, {s, t} = MinimalBy[{#, Complement[Range[n], #]}& /@ Subsets[Range[n]], Abs[Times @@ #[[1]] - Times @@ #[[2]]]&][[1]]; Times @@ s + Times @@ t];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 24}] (* Jean-François Alcover, May 06 2023 *)

a(n) <= A060696(n+1) = A076051(n) considering the interleaved partition b={2,4,6,..}, c={1,3,5, 7,...}. - R. J. Mathar, Jan 10 2007

a(n) = A200743(n) + A200744(n) = (A200744(n)^2 - A200743(n)^2) / A038667(n). - Max Alekseyev, Apr 08 2022

a(9)-a(13) from R. J. Mathar, Jan 10 2007

a(14)-a(26) from Ray Chandler, Feb 14 2007

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

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

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A061030 Factorial splitting: write n! = x*y*z with x

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A200744 Divide integers 1..n into two sets, minimizing the difference of their products. This sequence is the larger product.

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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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A038667 Minimal value of |product(A) - product(B)| where A and B are a partition of {1,2,3,...,n}.

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A127180 a(n) = smallest possible (product of b(k)'s + product of c(k)'s), where the positive integers <= n are partitioned somehow into {b(k)} and {c(k)}.

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Maple

Mathematica

A061030 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.