OFFSET
1,1
COMMENTS
Subset of A126706, numbers that are neither squarefree nor prime powers.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Annotated plot of b(n) = A126706(n), with n = 20*(y-1) + x at (x, -y), for x = 1..20 and y = 1..20, thus showing 400 terms. Terms in this sequence are colored black, those in A364999 in blue, in A364997 in green, and in A361098 in red.
Michael De Vlieger, Plot of b(n), with n = 120*(y-1) + x at (x, -y), for x = 1..120 and y = 1..120, thus showing 14400 terms, using the same color scheme as described immediately above.
Michael De Vlieger, Plot of b(n), with n = 1016*(y-1) + x at (x, -y), for x = 1..1016 and y = 1..1016, thus showing 1032256 terms. Terms in this sequence are colored black, else white. Demonstrates a strong quasiperiodic pattern approximately mod 169.
EXAMPLE
b(1) = 12 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6 = 30; both exceed 12, thus 12 is not in S.
b(2) = a(1) = 18 since p*r = 3*6 = 18 and q*r = 5*6 = 30. Indeed, 18 does not exceed 18 and 30 is larger than 18, hence 18 is in both S and T.
b(6) = 36 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6, and both do not exceed 36, therefore 36 is in S but not T.
b(7) = 40 is not in the sequence since p*r = 5*10 = 50 and q*r = 3*10 = 30. Though 50 > 40, 30 < 40, thus 40 is neither in S nor T, etc.
MATHEMATICA
Select[Select[Range[1500], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{k, f}, Function[{p, q, r}, And[p r <= k, q r > k]] @@ {f[[2, 1]], SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1], ! Divisible[k, #] &], Times @@ f[[All, 1]]}] @@ {#, FactorInteger[#]} &]
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Aug 16 2023
STATUS
approved