A346970 Smallest c which can be split into positive integers a,b with a+b=c, such that a*b*c is divisible by each of the first n primes.

2, 3, 5, 10, 21, 55, 182, 357, 1105, 2958, 16588, 51243, 106981, 608685, 3003455, 7497910, 52350909, 168539462, 1822961393, 3079378759

Offset

1,1

Comments

From David A. Corneth, Aug 10 2021: (Start)

Let k be a candidate value for a(n) and let g = gcd(c, primorial(n)) = gcd(c, A002110(n)). Let p be the largest prime that primorial(n)/g is divisible by. Then at least one of a and b is divisible by p.

So we can check multiples m of p < k and see if m*(k-m)*k is divisible by primorial(n).

If that is the case then k can be written as a + b where 0 < a,b < k and a*b*k is divisible by primorial(n) and a(n) = k when each of 1..k-1 do not have that property. (End)

From Kevin P. Thompson, Jun 30 2022 and Sep 07 2022: (Start)

n a(n) = c a b a * b * c

---- ---------- ---------- ----------- ---------------

1 2 1 1 prime(1)#

2 3 1 2 prime(2)#

3 5 2 3 prime(3)#

4 10 3 7 prime(4)#

5 21 10 11 prime(5)#

6 55 13 42 prime(6)#

7 182 17 165 prime(7)#

8 357 110 247 prime(8)#

9 1105 231 874 prime(9)#

10 2958 1463 1495 prime(10)#

11 16588 1615 14973 2 * prime(11)#

12 51243 16835 34408 4 * prime(12)#

13 106981 49335 57646 prime(13)#

14 608685 81548 527137 2 * prime(14)#

15 3003455 595034 2408421 7 * prime(15)#

16 7497910 2741301 4756609 3 * prime(16)#

17 52350909 2975429 49375480 4 * prime(17)#

18 168539462 55318167 113221295 9 * prime(18)#

19 1822961393 223726938 1599234455 83 * prime(19)#

20 3079378759 256763535 2822615224 4 * prime(20)# (End)

a(21) <= 111610913771, a(22) <= 111610913771 both via (a, b, c) = (54966571731, 56644342040, 111610913771). a(24) <= 8163073396289 via (a, b, c) = (133445799584, 8029627596705, 8163073396289). - David A. Corneth, Sep 07 2022

Links

Table of n, a(n) for n=1..20.

Formula

A002110(n)^(1/3) < a(n) <= A002110(n). - David A. Corneth, Aug 10 2021

a(2k) <= Product_{i=1..k} p_2i, a(2k+1) <= Product_{i=0..k} p_{2i+1} where p_i is the i-th prime. - Chai Wah Wu, Oct 14 2021

a(n) >= (4*A002110(n))^(1/3), strengthening David's result above. This inequality is strict for n > 1. - Charles R Greathouse IV, Jun 30 2022

Example

a(4) = 10 via 3+7=10 and 3*7*10 = 210 which is divisible by each of the first 4 primes, namely 2, 3, 5 and 7.
a(7) = 182 via 17+165=182. 17*165*182 = 510510 which is divisible by each of the first 7 primes, namely 2, 3, ..., 13, 17.

Mathematica

Do[Monitor[k=1; While[!Or@@And@@@(IntegerQ/@(#/Prime@Range@n)&/@Times@@@(Join[{k}, #]&/@IntegerPartitions[k, {2}])), k++]; Print@k, k], {n, 20}] (* Giorgos Kalogeropoulos, Aug 16 2021 *)

Prog

(PARI) a(n) = { if(n <= 3, return(prime(n))); P = vecprod(primes(n)); for(i = sqrtnint(P, 3) + 1, oo, if(iscanforA(n, i), return(i) ) ) }
iscanforA(n, k) = { my(g = gcd(k, P), step); if(k^2*g < P, return(0)); step = hpf(P/g); forstep(i = step, k, step, if((i*(k-i)*k)%P == 0,  return(1) ) ); 0 }
hpf(n) = my(f = factor(n)); f[#f~, 1] \\ David A. Corneth, Aug 10 2021
(Python)
from sympy import primorial, primefactors, gcd
def A346970(n):
    c, ps = 2, primorial(n)
    while True:
        m = ps//gcd(ps, c)
        if m == 1:
            return c
        p = max(primefactors(m))
        for a in range(p, c, p):
            if a*(c-a) % m == 0:
                return c
        c += 1 # Chai Wah Wu, Oct 13 2021

Crossrefs

Cf. A002110, A346971.

Sequence in context: A352879 A022861 A352818 * A001646 A103595 A293842

Adjacent sequences: A346967 A346968 A346969 * A346971 A346972 A346973

Keyword

nonn,more

Author

Steven M. Altschuld, Aug 09 2021

Extensions

a(12)-a(16) from David A. Corneth, Aug 10 2021

a(17)-a(18) from Kevin P. Thompson, Jun 30 2022

a(19)-a(20) from Kevin P. Thompson, Sep 07 2022

a(20) corrected by David A. Corneth, Sep 07 2022

Status

approved