A127180 -id:A127180 - cp's OEIS frontend

A200743 Divide integers 1..n into two sets, minimizing the difference of their products. This sequence is the smaller product.

1, 1, 2, 4, 10, 24, 70, 192, 576, 1890, 6300, 21600, 78624, 294840, 1140480, 4561920, 18849600, 79968000, 348566400, 1559376000, 7147140000, 33522128640, 160745472000, 787652812800, 3938264064000, 20080974513600, 104348244639744, 552160113120000, 2973491173785600, 16286186592000000, 90678987245246400

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) = 78624.

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

Cf. A060776, A038667, A127180, A200744.

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])); 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]]; Min[Times @@ s, Times @@ t]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 25}] (* Jean-François Alcover, Nov 03 2020 *)

from itertools import combinations
def prod(l):
    t=1
    for x in l:
        t *= x
    return t
def a200743(n):
    nums = list(range(1,n+1))
    widths = combinations(nums,n//2)
    dimensions = [(prod(width),prod(x for x in nums if x not in width)) for width in widths]
    best = min(dimensions,key=lambda x:max(*x)-min(*x))
    return min(best)
# Christian Perfect, Feb 04 2015

from math import prod, factorial
from itertools import combinations
def A200743(n):
    m = factorial(n)
    return min((abs((p:=prod(d))-m//p),min(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) - A200744(n) = A200744(n) - A038667(n) = (A127180(n) - A038667(n)) / 2. - Max Alekseyev, Jun 18 2022

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

a(31) from Michael S. Branicky, May 21 2021

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

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

A182987 Least a + b such that a*b = A002110(n), the product of the first n primes, where a, b are positive integers.

Original entry on oeis.org

2, 3, 5, 11, 29, 97, 347, 1429, 6229, 29873, 160879, 895681, 5448239, 34885673, 228759799, 1568299433, 11417382973, 87698582693, 684947829299, 5606539600699, 47241542381273, 403631914511993, 3587558929043927, 32684217334524347, 308342289648328511, 3036819365023723883

Offset: 0

Risto Kauppila, Feb 06 2011

nonn

Original definition (not applicable for n = 0 and 1, but equivalent for n >= 2):
Let p(S) be product of integers in S. a(n) is minimum of p(S_1) + p(S_2) over all partitions of first n primes into sets S_1 and S_2.
Also: Least integer such that a(n)^2 - 4*A002110(n) is a square. - David Broadhurst, Sep 20 2011
The integers a,b are the two median divisors of primorial(n), a = A060795(n) = A060775(A002110(n)) and b = A060796(n) = A033677(A002110(n)). (For n = 0, a = b = 1 of course.) - M. F. Hasler, Sep 20 2011

			a(3) = 11 = min{ 2*3 + 5 = 11, 2*5 + 3 = 13, 3*5 + 2 = 17 }
Or, a(3) = 11 = min { 1+30, 2+15, 3+10, 5+6 } because A002110(3) = 2*3*5 = 30 = 2*15 = 3*10 = 5*6.

Max Alekseyev, Table of n, a(n) for n = 0..70
David Broadhurst, Re: adding to prime number [primes in A182987], primenumbers group, Sep 20 2011.
David Broadhurst and others, Adding to prime number, digest of 28 messages in primenumbers Yahoo group, Sep 19, 2011 - Sep 22, 2011.

Cf. A173631, A060795, A060796, A061060, A127180.

Cf. A000196 (integer sqrt), A002110 (primorial), A010052 (is_square).

a[0] = 2; a[n_] := Module[{m = Times @@ Prime[Range[n]]}, For[an = 2 Floor[Sqrt[m]] + 1, an <= m + 2, an += 2, If[IntegerQ[Sqrt[an^2 - 4 m]], Return[an]]]]; Table[an = a[n]; Print[an]; an, {n, 0, 25}] (* Jean-François Alcover, Oct 20 2016, adapted from PARI *)

A182987(n)={if(n,vecsum(divisors(vecprod(primes(n)))[2^(n-1)..2^(n-1)+1]),2)}  \\ Needs stack size >= 2^(n+6). - M. F. Hasler, Sep 20 2011, edited Mar 24 2022

A182987(n)={ n||return(2); my(m=prod(k=1,n,prime(k))); forstep(a=2*sqrtint(m)+1,m+2,2, issquare(a^2-4*m) & return(a)) }  \\ Slow for n >= 28, but needs not much memory. - M. F. Hasler, following an idea from David Broadhurst, Sep 20 2011

def A182987(n): # becomes slow for n >= 28, but not slower than memory-hungry
   # sum(divisors(primorial(n))[2**(n-1)-1:2**(n-1)+1]) if n else 2
   "Compute A182987(n) = sum of the two central divisors of primorial(n)."
   if n < 2: return n+2
   from math import isqrt # = A000196
   from sympy import primorial # = A002110
   from sympy.ntheory.primetest import is_square # = A010052
   m = primorial(n)*4; a = isqrt(m)|1  ### ceil(sqrt(m))**2 < m  for n = 26..27 !!
   while True:
      if is_square(a*a - m): return a
      a += 2
# M. F. Hasler, Mar 21 2022

a(n) = A060795(n) + A060796(n). - M. F. Hasler, Sep 20 2011

Conjecture: Limit_{N->oo} (Sum_{n=1..N} log(a(n)-2*sqrt(prime(n)#))) / (Sum_{n=1..N} prime(n)) = 2/e - 1/2 (i.e., A135002 - 1/2). - Alain Rocchelli, Nov 30 2023

First term and example corrected, as empty sets have product 1, by Phil Carmody, Sep 20 2011
Simpler definition and extension to n = 0 by M. F. Hasler, Sep 20 2011
b-file extended to a(59) with results from R. Gerbicz (cf. 2nd Broadhurst link) by M. F. Hasler, Mar 22 2022
a(60)-a(70) in b-file from Max Alekseyev, Apr 20 2022

A127181 a(1)=a(2)=1. a(n) = smallest possible (product of b(k)'s + product of c(k)'s), where the sequence's terms a(1) through a(n-1) are partitioned somehow into {b(k)} and {c(k)}.

Original entry on oeis.org

1, 1, 2, 3, 5, 11, 37, 221, 3361, 190777, 83199527, 760382931109, 662056785094857629, 538451433632092674800570837, 12495147956629620251492228703104952798089, 1397663545252630798358314360015943050984074671707253231083973

Offset: 1

Leroy Quet, Jan 07 2007

nonn

Every term of the sequence is coprime to every other term.

			By partitioning (a(1),a(2),...a(7)) = (1,1,2,3,5,11,37) into {b(k)} and {c(k)} so that {b(k)} = (1,2,5,11) and {c(k)} = (1,3,37), then (product of b(k)'s + product of c(k)'s) is minimized. Therefore a(8) = 1*2*5*11 + 1*3*37 = 221.

Max Alekseyev, Table of n, a(n) for n = 1..30

Cf. A127180.

Nest[ Module[ {prod=Times@@#1}, Append[ #,Min[ #+prod/#&/@Times@@@Union[ Subsets[ # ] ] ] ] ]&,{1,1,2,3},12 ] (* Peter Pein (petsie(AT)dordos.net), Jan 07 2007 *)

a(10)-a(15) from Peter Pein (petsie(AT)dordos.net), Jan 07 2007

a(16)-a(30) from Max Alekseyev, Apr 08 2022

cp's OEIS Frontend

A200743 Divide integers 1..n into two sets, minimizing the difference of their products. This sequence is the smaller product.

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

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A182987 Least a + b such that a*b = A002110(n), the product of the first n primes, where a, b are positive integers.

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A127181 a(1)=a(2)=1. a(n) = smallest possible (product of b(k)'s + product of c(k)'s), where the sequence's terms a(1) through a(n-1) are partitioned somehow into {b(k)} and {c(k)}.

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