



36, 48, 50, 54, 72, 75, 80, 96, 98, 100, 108, 112, 135, 144, 147, 160, 162, 189, 192, 196, 200, 216, 224, 225, 240, 242, 245, 250, 252, 270, 288, 294, 300, 320, 324, 336, 338, 350, 352, 360, 363, 375, 378, 384, 392, 396, 400, 405, 416, 432, 441, 448, 450, 468, 480, 484, 486, 490, 500, 504, 507, 525
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OFFSET

1,1


COMMENTS

Numbers k that are neither prime powers nor squarefree, such that rad(k) * A053669(k) < k and k/rad(k) >= A119288(k), where rad(k) = A007947(k).
Subset of A126706. All terms are neither prime powers nor squarefree.


LINKS

Michael De Vlieger, 1016 X 1016 pixel bitmap read left to right in rows, then top to bottom where the kth pixel is black if A126706(k) is in this sequence, else white (1032256 pixels total).


EXAMPLE

For prime power n = p^e > 4, e > 0, A360543(n) = p^(e1)  e, but A360480(n) = A361235(n) = A355432(n) = 0, since the other sequences require omega(n) > 1.
For squarefree composite n, A360480(n) >= 1 and A361235(n) >= 1 (the latter for n > 6), but A360543(n) = A355432(n) = 0, since the other sequences require at least 1 prime power factor p^e  n with e > 0.
For n = 18, A360480(n) =  {10, 14, 15}  = 3,
Therefore 18 is not in the sequence.
For n = 36, A360480(n) =  {10, 14, 15, 20, 21, 22, 26, 28, 33, 34}  = 10,
A361235(n) =  {8, 16, 27, 32}  = 4,
Therefore 36 is the smallest term in the sequence.
Table pertaining to the first 12 terms:
n  a + b = c  d + e = f  g + tau + phi  1 = n

36  10 + 1 = 11  4 + 1 = 5  16 + 9 + 12  1 = 36
48  16 + 2 = 18  3 + 2 = 5  23 + 10 + 16  1 = 48
50  18 + 1 = 19  4 + 2 = 6  25 + 6 + 20  1 = 50
54  19 + 2 = 21  4 + 4 = 8  29 + 8 + 18  1 = 54
72  27 + 4 = 31  4 + 2 = 6  37 + 12 + 24  1 = 72
75  25 + 2 = 27  2 + 1 = 3  30 + 6 + 40  1 = 75
80  32 + 3 = 35  3 + 1 = 4  39 + 10 + 32  1 = 80
96  38 + 7 = 45  4 + 4 = 8  53 + 12 + 32  1 = 96
98  41 + 3 = 44  5 + 2 = 7  51 + 6 + 42  1 = 98
100  42 + 4 = 46  4 + 2 = 6  52 + 9 + 40  1 = 100
108  44 + 8 = 52  5 + 4 = 9  61 + 12 + 36  1 = 108
112  48 + 3 = 51  3 + 1 = 4  55 + 10 + 48  1 = 112


MATHEMATICA

nn = 2^16;
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, n/#1 >= #2] & @@ {Times @@ #, #[[2]]} &@
FactorInteger[n][[All, 1]], Sow[n]], {j, Length[s]}]][[1, 1]]


CROSSREFS

Cf. A000005, A000010, A001694, A002182, A007947, A045763, A053669, A119288, A126706, A168263, A243822, A243823, A286708, A303606, A355432, A359280, A360480, A360543, A361235.


KEYWORD

nonn


AUTHOR



STATUS

approved



