A157940 -id:A157940 - cp's OEIS frontend

A157938 Numbers n divisible by the least prime >= sqrt(n) but not by the largest prime <= sqrt(n).

10, 20, 28, 42, 55, 66, 88, 99, 110, 130, 156, 170, 187, 204, 238, 255, 272, 304, 342, 368, 391, 414, 460, 483, 506, 551, 580, 609, 638, 696, 725, 754, 783, 812, 868, 930, 962, 999, 1036, 1073, 1110, 1184, 1221, 1258, 1295, 1332, 1394, 1435, 1476, 1558

Offset: 1

M. F. Hasler, Mar 10 2009

nonn

Also: Numbers n divisible by the least prime >= sqrt(n) which are not in A001248 (primes squared) or A006094 (product of two consecutive primes). A subsequence of A157937.

			a(1)=10 and a(2)=20 are divisible by 5 = nextprime(sqrt(10)) = nextprime(sqrt(20)) and neither a prime squared (as are 4 and 9) nor product of consecutive primes (as are 6 and 15).
5,7,8 are not in this sequence, since not a multiple of 3=nextprime(sqrt(5))=nextprime(sqrt(8)).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A157940.

dpQ[n_]:=Module[{srn=Sqrt[n],a,b},a=If[PrimeQ[srn],srn,NextPrime[ srn]];b=If[PrimeQ[srn],srn,NextPrime[srn,-1]]; Divisible[n,a]&& !Divisible[ n,b]]; Select[Range[2000],dpQ] (* Harvey P. Dale, Oct 10 2011 *)

for( n=5,1999, n % nextprime(sqrtint(n-1)+1) & next; n % precprime(sqrtint(n)) & print1(n","))

A157938 = A157936 \ A157942 = A157937 \ A006094, where A157937 = A157936 \ A001248.

A157939 Numbers n divisible by precprime(sqrt(n)) or nextprime(sqrt(n)) but not both, where precprime=A007917, nextprime=A007918.

Original entry on oeis.org

8, 10, 12, 18, 20, 21, 24, 28, 30, 40, 42, 45, 55, 56, 63, 66, 70, 84, 88, 91, 98, 99, 105, 110, 112, 119, 130, 132, 154, 156, 165, 170, 182, 187, 195, 204, 208, 234, 238, 247, 255, 260, 272, 273, 286, 304, 306, 340, 342, 357, 368, 380, 391, 399, 414, 418, 456

Offset: 1

M. F. Hasler, Mar 10 2009

nonn

Equal to the union of its disjoint subsequences A157938 and A157940.

			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)).
a(2) = 10 is divisible by 5=nextprime(sqrt(10)) but not by 3=precprime(sqrt(8)).
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)).

Project Euler, Problem 234: Semidivisible numbers, Feb 28 2009

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 *)

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 */

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.

cp's OEIS Frontend

A157938 Numbers n divisible by the least prime >= sqrt(n) but not by the largest prime <= sqrt(n).

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