A334468 List of distinct values of n + A217287(n).

4, 6, 8, 12, 15, 16, 18, 24, 27, 30, 32, 36, 40, 45, 48, 54, 60, 63, 64, 70, 72, 75, 80, 81, 90, 96, 100, 105, 108, 120, 125, 128, 135, 140, 144, 150, 160, 162, 168, 175, 180, 189, 192, 200, 210, 216, 224, 225, 234, 240, 243, 250, 256, 270, 280, 288, 294, 300

Offset: 1

Author

Michael De Vlieger, May 02 2020

Keywords

• nonn

Comments

This sequence is a list of primitive least m > n whose distinct prime factors p are not a subset of those prime factors p found in the range n..(m - 1), i.e., the smallest A217287(n)-smooth number m > n. These numbers serve as "obstructions" that end or break the chains described at A217287.

The number (a(n) - 1) can be found in at least one row of A217438. In other words, this sequence includes any number T(n, A217287(n)) + 1 where T(n, k) is the irregular triangle described at A217438.

Examples

			Start with n = 1, the empty product. Incrementing n and storing the distinct prime factors each time, we encounter 2, which does not divide any previous number n. Therefore we proceed to n = 3, which is prime and its distinct prime divisor again does not divide any previous number. Finally, at 4, we have the distinct prime divisor 2, since 2 divides the product of the previous range {1, 2, 3}, we end the chain. Therefore 4 is the first term of this sequence.
We list row n of A217438 below, starting with n aligned in columns:
1  2  3
   2  3
      3  4  5
         4  5  6  7
            5  6  7
               6  7
                  7  8  9  10  11
                     8  9  10  11
                        9  10  11
                           10  11  12  13  14
                               11  12  13  14  15
                                   12  13  14  15
                                       13  14  15
                                           14  15
                                               ...
Adding 1 to the last numbers seen in all the rows and considering each value only once, we generate the sequence {4, 6, 8, 12, 15, 16, ...}
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Plot (x,y) of x in rows 1 <= y <= 4096 of A217438 in gray, with the single pixel m (in this sequence) that exceeds the largest term of A217438 in red.
• Michael De Vlieger, Analysis of prime decompositions of terms in this sequence.

Crossrefs

Cf. A000918, A217287, A217438, A334469.

Programs

Mathematica

Block[{nn = 2^9, r}, r = Array[If[# == 1, 0, Total[2^(PrimePi /@ FactorInteger[#][[All, 1]] - 1)]] &, nn + Ceiling@ Sqrt@ nn]; Union@ Array[Block[{k = # + 1, s = r[[#]]}, While[UnsameQ[s, Set[s, BitOr[s, r[[k]] ] ] ], k++]; k] &, nn] ]

Formula

a(n) > n + 2 for all n.

a(m) = m + 2 for m = 2^k - 2 and k > 1, since m is even and 2^k only has the distinct prime divisor 2. Therefore 2^k for k > 1 is in this sequence.