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A334468
List of distinct values of n + A217287(n).
3
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
1,1
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.
LINKS
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.
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.
EXAMPLE
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, ...}
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] ]
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael De Vlieger, May 02 2020
STATUS
approved