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

1,1

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).

Numbers k such that A360480(k), A360543(k), A361235(k), and A355432(k) are positive.

Subset of A126706. All terms are neither prime powers nor squarefree.

From Michael De Vlieger, Aug 03 2023: (Start)

Superset of A286708 = A001694 \ {{1} U A246547}, which in turn is a superset of A303606. We may write k in A286708 as m*rad(k)^2, m >= 1. Since omega(k) > 1, it is clear both k/rad(k) > A053669(k) and k/rad(k) >= A119288(k). Also superset of A359280 = A286708 \ A303606.

This sequence contains {A002182 \ A168263}. (End)

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 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
  ------------------------------------------------------
   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, A355432, A360480, A360543, A361235.

Subsets: A095990\{18}, A286708, A303606, A359280.

Sequence in context: A083674 A160063 A260927 * A291713 A370266 A287840

Adjacent sequences: A361095 A361096 A361097 * A361099 A361100 A361101

Keyword

nonn

Author

Michael De Vlieger, Mar 15 2023

Status

approved