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.
From Michael De Vlieger, Aug 03 2023: (Start)
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Diagram showing k = 1..n for n = 1..54 in blue for k counted by A360480(n), in green for k counted by A360543(n), in gold for k counted by A361235(n), and in magenta for k counted by A355432(n). Red dots indicate k | n such that k > 1, while gray dots indicate gcd(k, n) = 1.
Michael De Vlieger, 1016 X 1016 pixel bitmap read left to right in rows, then top to bottom where the k-th 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^(e-1) - 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,
A360543(n) = | {} | = 0,
A361235(n) = | {4, 8, 16} | = 3,
A355432(n) = | {12} | = 1.
Therefore 18 is not in the sequence.
For n = 36, A360480(n) = | {10, 14, 15, 20, 21, 22, 26, 28, 33, 34} | = 10,
A360543(n) = | {30} | = 1,
A361235(n) = | {8, 16, 27, 32} | = 4,
A355432(n) = | {24} | = 1.
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
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Mar 15 2023
STATUS
approved