A226891 Least k such that k(k+1)(k+2)(k+3) is divisible by prime(n)#.

1, 2, 4, 19, 63, 153, 1273, 2090, 19227, 266133, 868868, 10631543, 264365332, 662822809, 129102309125

Offset: 2

Author

Charles R Greathouse IV, Jun 20 2013

Keywords

• nonn
• more

Comments

Essentially indices of records in A053672.

Examples

			63 is in the sequence because {63, 64, 65, 66} are divisible by 2, 3, 5, 7, 11, and 13; no smaller number is divisible by all of these primes.
		

Crossrefs

Cf. A053672, A078638, A002110.

Programs

PARI

a(n)=if(n<3,return(1));my(p=prime(n),P=prod(i=1,n-1,prime(i))/6, t=sqrtnint(24*p^2*P,4)+1); forstep(k=max(t\p,1)*p-3,P+2,[1,1,1,p-3], if(gcd(P, (k+3)*(k+2)*(k^2+k))==P,return(k)))

Formula

Let p be the n-th prime, then (4p#)^(1/4) - 2 < a(n) < p#; in particular, a(n) >> exp(p/4).

Extensions

a(16) from Charles R Greathouse IV, Jun 21 2013