A246463 a(n) = min(p +- q) > 1 with p*q being equal to the n-th primorial (A002110).

5, 7, 11, 13, 17, 107, 41, 157, 1811, 1579, 18859, 95533, 17659, 1995293, 208303, 2396687, 58513111, 299808329, 3952306763, 341777053, 115405393057, 437621467859, 1009861675153, 6660853109087, 29075165225531, 418895584426457, 2371362636817019, 6889206780487667, 5258351738694673

Offset: 2

Author

Robert G. Wilson v, Aug 26 2014

Keywords

• nonn

Comments

All terms must be odd since 2 is only represented once in the factorization of any primorial. The two divisors of different parity, p and q, must straddle the square root of the primorial.

Examples

			a(7) = 41 since the middle four divisors of 7# or 2*3*5*7*11*13*17 = 510510 are 663, 714, 715 and 770. Because the middle two only differ by 1, the next pair, 663 and 770 are used and their difference is 107.
a(8) = 41 since the middle two divisors of 8# are 3094 and 3135 which have a difference of 41.
		

Crossrefs

Cf. A003681, A002110.

Programs

Mathematica

f[n_] := Block[{k = mx = 1, fi = Prime@ Range@ n, prod = Fold[Times, 1, Prime@ Range@ n], sqrt, tms}, sqrt = Floor@ Sqrt@ prod; While[k < 2^n, tms = Times @@ (fi^IntegerDigits[k, 2, n]); If[mx < tms < sqrt, mx = tms]; k++]; prod/mx - mx]; Array[f, 30, 2]

PARI

a(n)=my(P=prod(i=1,n,prime(i)));forstep(k=sqrtint(P),1,-1,if(P%k==0 && P/k-k>1, return(P/k-k))) \\ Charles R Greathouse IV, Aug 31 2014

PARI

a(n)=my(P=prod(i=1,n,prime(i)),t); fordiv(P,d, if(P/d-d>1, t=P/d-d, return(t))) \\ Charles R Greathouse IV, Aug 31 2014