A061058 Duplicate of A060795.
1, 2, 5, 14, 42, 165, 714, 3094, 14858, 79534, 447051, 2714690, 17395070
Offset: 1
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.
For n = 1, two distinct fractions can be written with the first prime number, namely 1/2 and 2. Of the two, 1/2 is nearer to 1. 1/2 has its 2 below the fraction bar, so its binary encoding is 0, which yields a(1) = 0. For n = 2, four distinct fractions can be written with the first two prime numbers, namely 1/6, 2/3, 3/2 and 6. 2/3 is the nearest to 1. 2/3 has its 2 above the fraction bar and its 3 below, so its encoding is 01, which yields a(2) = 1.
package oeis;
public class BinaryEncodedBestPrimeSetup {
// Brute force implementation... Can it be improved?
public static int PRIME[] = { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, /* to be continued */ };
public static void main(String args[]) {
int nMax = PRIME.length; // number of terms of the sequence
for (int n = 1; n <= nMax; n ++) {
if (n > 1) {
System.out.print(", ");
}
System.out.print(u(n));
}
}
private static int u(int n) {
double bestMul = 0.0;
int bestSetup = -1;
int s = 0; // binary-encoded setup number
for (s = 0; s < (1 << n); s ++) {
double mul = 1.0;
int i = 0; // prime number #
for (i = 0; i < n; i ++) {
if ((s & (1 << i)) != 0) {
mul *= PRIME[i]; // 1 = above fraction bar
} else {
mul /= PRIME[i]; // 0 = below fraction bar
}
}
if (mul < 1.0) {
if (mul > bestMul) {
bestMul = mul;
bestSetup = s;
}
}
}
return bestSetup;
}
}
{0}~Join~Table[Function[p, FromDigits[#, 2] &@ Reverse@ MapAt[# + 1 &, ConstantArray[0, n], Partition[#, 1, 1]] &@ PrimePi@ FactorInteger[Numerator@ #][[All, 1]] &@ Max@ Select[Map[p/#^2 &, Divisors@ p], # < 1 &]][Times @@ Prime@ Range@ n], {n, 2, 23}] (* Michael De Vlieger, Oct 19 2016 *)
a060795(n) = my(m=prod(i=1, n, prime(i))); divisors(m)[numdiv(m)\2];
a(n) = {my(a95 = a060795(n)); v = vector(n, i, (a95 % prime(i))==0); subst(Polrev(v), x, 2); } \\ Michel Marcus, Dec 03 2016
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]
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
For n = 6, 6! = 720 = 8*9*10, so x=8, y=9, z=10.
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).
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(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
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 *)
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.
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
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