A060796 -id:A060796 - cp's OEIS frontend

A061059 Duplicate of A060796.

2, 3, 6, 15, 55, 182, 715, 3135, 15015, 81345, 448630, 2733549

Offset: 1

dead

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

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

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

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.

cp's OEIS Frontend

A061059 Duplicate of A060796.

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

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Maple

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

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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.