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

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

1,1

Comments

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

Links

Table of n, a(n) for n=1..57.

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

Formula

Equals 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.

Example

For k=1,2,3, precprime(sqrt(k)) 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)).
k=4,6,9,... are excluded since divisible by both precprime(sqrt(k)) and nextprime(sqrt(k)). (Note that precprime=A007917 and nextprime=A007918 are defined using weak inequalities.)
k=5,7,11,13,14,... are excluded since not divisible by precprime(sqrt(k)) nor by nextprime(sqrt(k)).

Mathematica

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 *)

Prog

(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 */

Crossrefs

Sequence in context: A249628 A352169 A158273 * A208154 A096281 A114873

Adjacent sequences: A157936 A157937 A157938 * A157940 A157941 A157942

Keyword

nonn

Author

M. F. Hasler, Mar 10 2009

Status

approved