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%I #8 Feb 21 2020 20:12:39
%S 8,10,12,18,20,21,24,28,30,40,42,45,55,56,63,66,70,84,88,91,98,99,105,
%T 110,112,119,130,132,154,156,165,170,182,187,195,204,208,234,238,247,
%U 255,260,272,273,286,304,306,340,342,357,368,380,391,399,414,418,456
%N Numbers n divisible by precprime(sqrt(n)) or nextprime(sqrt(n)) but not both, where precprime=A007917, nextprime=A007918.
%C Equal to the union of its disjoint subsequences A157938 and A157940.
%H Project Euler, <a href="https://projecteuler.net/index.php?section=problems&id=234">Problem 234: Semidivisible numbers</a>, Feb 28 2009
%F A157939 = A157938 union A157940 = A157937 /\ A157941 = A157936 /\ A157942, where A /\ B = (A u B) \ (A n B) = (A \ B) u (B \ A) is the symmetric difference; A157937 intersect A157941 = A006094 = (A157936 intersect A157942) \ A001248.
%e For n=1,2,3, precprime(sqrt(n)) is undefined, so these are not considered here. a(1) = 8 is divisible by 2=precprime(sqrt(8)) but not by 3=nextprime(sqrt(8)).
%e a(2) = 10 is divisible by 5=nextprime(sqrt(10)) but not by 3=precprime(sqrt(8)).
%e n=4,6,9,... are excluded since divisible by both precprime(sqrt(n)) and nextprime(sqrt(n)). (Note that precprime=A007917 and nextprime=A007918 are defined using weak inequalities.) n=5,7,11,13 but also 14 are excluded since not divisible by precprime(sqrt(n)) nor by nextprime(sqrt(n)).
%t ndQ[n_]:=Module[{s=Sqrt[n]},Total[Boole[{Divisible[n,NextPrime[ s]], Divisible[ n, NextPrime[ s,-1]]}]]==1]; Select[Range[5,500],ndQ] (* _Harvey P. Dale_, Mar 19 2019 *)
%o (PARI) for( n=4,999, !(n % nextprime(sqrtint(n-1)+1)) != !(n % precprime(sqrtint(n))) & print1(n",")) /* sqrtint(n-1)+1 avoids rounding errors but can be replaced by sqrt(n) for small n */
%K nonn
%O 1,1
%A _M. F. Hasler_, Mar 10 2009
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