A361098 Intersection of A360765 and A360768.
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
Offset: 1
Keywords
Examples
For prime p, A360480(p) = A360543(p) = A361235(p) = A355432(p) = 0, since k < p is coprime to p.
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:
Key: a = A360480, b = A360543, c = A243823; d = A361235, e = A355432, f = A243822;
g = A046753 = f + c, tau = A000005, phi = A000010.
n | a + b = c | d + e = f | g + tau + phi - 1 = n
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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
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).
Crossrefs
Programs
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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]]
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