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

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

Author

Risto Kauppila, Feb 06 2011

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• 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.

Crossrefs

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

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

Programs

Mathematica

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 *)

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

PARI

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

Python

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

Formula

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

Extensions

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