cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-17 of 17 results.

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

Views

Author

Ed Pegg Jr, May 28 2001

Keywords

Comments

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

Examples

			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).

Links

Crossrefs

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

Programs

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

Formula

a(n) = A060777(n) - A060776(n).
a(n) = A056737(A000142(n)). - Pontus von Brömssen, Jul 15 2023

Extensions

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

Views

Author

Ed Pegg Jr, May 28 2001

Keywords

Examples

			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

Links

Crossrefs

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

Programs

  • Maple
    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]
  • Mathematica
    (* 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 *)

Formula

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

Extensions

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! = x*y*z 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

Views

Author

Max Alekseyev, Jun 23 2022

Keywords

Comments

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.

Links

Crossrefs

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

A355190 Factorial splitting: write n! = x*y*z 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

Views

Author

Max Alekseyev, Jun 23 2022

Keywords

Links

Crossrefs

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

A355191 Factorial splitting: write n! = x*y*z 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

Views

Author

Max Alekseyev, Jun 23 2022

Keywords

Comments

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.

Links

Crossrefs

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

A355192 Factorial splitting: write n! = x*y*z with x <= y <= z and minimal z-x; sequence gives value of z-x.

Original entry on oeis.org

0, 0, 1, 2, 2, 2, 2, 6, 4, 2, 22, 28, 42, 72, 112, 288, 420, 720, 1568, 1920, 3512, 16560, 19686, 16028, 71280, 182160, 184320, 552960, 2925648, 4885160, 12080640, 32861900, 71559680, 77631750, 217165824, 604653336, 368858880, 4069377144, 7919402400, 17537715360, 87352688640, 127718553600
Offset: 0

Views

Author

Max Alekseyev, Jun 23 2022

Keywords

Comments

a(n) <= A061033(n).
n=61 gives the smallest example where the value of x is not maximal (cf. A061030) and the value of z is not minimal.
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.

Links

Crossrefs

Cf. A061030, A061031, A061032, A061033, A060776, A060777, A060795, A060796, A200743, A200744, A355189, A355190, A355191.

A060794 Difference between upper and lower central divisors of n.

Original entry on oeis.org

1, 2, 1, 4, 1, 6, 2, 2, 3, 10, 1, 12, 5, 2, 2, 16, 3, 18, 1, 4, 9, 22, 2, 4, 11, 6, 3, 28, 1, 30, 4, 8, 15, 2, 2, 36, 17, 10, 3, 40, 1, 42, 7, 4, 21, 46, 2, 6, 5, 14, 9, 52, 3, 6, 1, 16, 27, 58, 4, 60, 29, 2, 4, 8, 5, 66, 13, 20, 3, 70, 1, 72, 35, 10, 15, 4, 7, 78, 2, 6, 39, 82, 5, 12, 41, 26
Offset: 2

Views

Author

Labos Elemer, Apr 27 2001

Keywords

Examples

			Difference between upper and lower central divisors may be small or relatively large. So neither A060775 nor A033677 are always good central divisors as to their magnitude. n=182,D={1,2,7,13,14,26,91,182}; central divisors={13,14}, difference=1. n=254, D={1,2,127,254}, central divisors={2,127}, a(254)=125. n=p, D={1,p}. Here the central divisors are also marginal ones: a(p)=p-1.

Links

Crossrefs

a(n) = A033677(n) - A060775(n).
Cf. A060775, A060776, A060777, A033677, A000196.

Programs

  • Mathematica
    a(n)=Part[Divisors[n], 1+cd[n]]-Part[Divisors[n], cd[n]], where cd[x_] := cd[x_] := Floor[DivisorSigma[0, x]/2]
  • PARI
    a(n)={my(d=divisors(n)); if(n>1, d[1 + #d\2] - d[#d\2], 0)} \\ Harry J. Smith, Jul 12 2009
Previous Showing 11-17 of 17 results.

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