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%I #14 Mar 28 2015 17:28:19
%S 5,7,11,13,17,107,41,157,1811,1579,18859,95533,17659,1995293,208303,
%T 2396687,58513111,299808329,3952306763,341777053,115405393057,
%U 437621467859,1009861675153,6660853109087,29075165225531,418895584426457,2371362636817019,6889206780487667,5258351738694673
%N a(n) = min(p +- q) > 1 with p*q being equal to the n-th primorial (A002110).
%C 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.
%e 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.
%e a(8) = 41 since the middle two divisors of 8# are 3094 and 3135 which have a difference of 41.
%t 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]
%o (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
%o (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
%Y Cf. A003681, A002110.
%K nonn
%O 2,1
%A _Robert G. Wilson v_, Aug 26 2014
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