A128281 a(n) is the least product of n distinct odd primes m=p_1*p_2*...*p_n, such that (d+m/d)/2 are all primes for each d dividing m.

3, 21, 105, 1365, 884037

Offset: 1

Author

Kok Seng Chua (chuakokseng(AT)hotmail.com), Mar 05 2007

Keywords

• nonn
• hard
• more

Comments

From Iain Fox, Aug 26 2020: (Start)

a(6) > 10^9 if it exists.

All terms are members of A076274 since the definition requires that (1+m)/2 be prime.

The number of prime factors of m congruent to 3 (mod 4) must be even except for n=1.

(End)

a(6) > 2*10^11 if it exists. - David A. Corneth, Aug 27 2020

a(n) >= A070826(n+1) by definition of the sequence. - Iain Fox, Aug 28 2020

Examples

			105=3*5*7, (3*5*7+1)/2=53, (3+5*7)/2=19, (5+3*7)/2=13, (7+3*5)/2=11 are all primes and 105 is the least such number which is the product of 3 primes, so a(3)=3.
		

Crossrefs

Subsequence of A076274.

Lower bound: A070826.

Cf. A002145, A128276, A128282, A128283, A128284, A128285, A128286, A168352.

Programs

PARI

a(n)=if(n==1, return(3)); my(p=prod(k=1, n, prime(k+1))); forstep(m=p+if(p%4-1, 2), +oo, 4, if(bigomega(m)==n && omega(m)==n, fordiv(m, d, if(!isprime((d+m/d)/2), next(2))); return(m))) \\ Iain Fox, Aug 27 2020

Extensions

Definition corrected by Iain Fox, Aug 25 2020