OFFSET
1,1
COMMENTS
Subset of A126706, the set of numbers k neither prime powers nor squarefree, i.e., k such that A001222(k) > A001221(k) > 1.
Let p = A119288(k) be the second smallest prime factor of k. Let q = A053669(k) be the smallest prime that does not divide k. Let r = rad(k) = A007947(k) be the squarefree kernel of k. Define sequence S = A361098 = {k : Omega(k) > omega(k) > 1, q*r < k, p*r <= k} = A361098.
Sequence T = A286708 represents numbers in A001694 that are not prime powers. Numbers k in T are such that k = m*r^2, m >= 1, by definition. Since we may rewrite q*r < k instead as q*r < m*r^2, it is clear since omega(r) > 1, that q < r. Further, we may rewrite p*r <= k instead as p*r <= m*r^2, and since p | r, p < r as omega(r) > 1, we see that S contains T.
This sequence gives k that are in S but not in T.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
EXAMPLE
Let B = A126706.
B(1) = 12 is not in the sequence since 3*6 > 12.
B(2) = 18 is not in the sequence, since, though 3*6 = 18, 5*6 > 18.
B(6) = S(1) = 36 is not in the sequence since, though 3*6 < 36 and 5*6 < 36, rad(36)^2 = 6^2 | 36, hence B(6) = T(1).
B(10) = S(2) = a(1) = 48 is in the sequence since rad(48) = 6, and 6^2 does not divide 48.
B(11) = S(3) = a(2) = 50 is in the sequence since rad(50) = 10, and 10^2 does not divide 50, etc.
MATHEMATICA
nn = 2^10; a053669[n_] := If[OddQ[n], 2, p = 2; While[Divisible[n, p], p = NextPrime[p]]; p]; s = Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &]; Reap[Do[n = s[[j]]; If[And[#1*a053669[n] < n, #1*#2 <= n, ! Divisible[n, #1^2]] & @@ {Times @@ #, #[[2]]} &@ FactorInteger[n][[All, 1]], Sow[n]], {j, Length[s]}] ][[-1, -1]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Aug 03 2023
STATUS
approved