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A364262
a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared such that a(n-1) + a(n) has the same number of distinct prime factors as a(n-1) * a(n).
4
1, 2, 10, 4, 6, 8, 7, 3, 11, 9, 5, 13, 23, 16, 12, 24, 27, 17, 19, 25, 15, 51, 33, 37, 31, 32, 14, 46, 20, 22, 38, 28, 42, 18, 36, 30, 40, 26, 34, 44, 58, 29, 43, 35, 55, 47, 41, 53, 52, 50, 60, 45, 21, 39, 63, 49, 59, 64, 48, 57, 69, 61, 65, 67, 79, 73, 71, 81, 54, 56, 70, 80, 74, 76, 62, 68, 72
OFFSET
1,2
COMMENTS
The terms are concentrated along the line a(n) = n, resulting in seven-hundred six fixed points in the first 50000 terms. These begin 1, 2, 4, 7, 19, 43, 50, 134, ... . See the linked image. In the same range the smallest unseen number is 46410, suggesting all numbers will eventually appear.
LINKS
Scott R. Shannon, Image of the first 50000 terms. The green line is a(n) = n.
EXAMPLE
a(2) = 2 as a(1) + 2 = 1 + 2 = 3 while a(1) * 2 = 1 * 2 = 2, both of which have one distinct prime factor.
a(3) = 10 as a(2) + 10 = 2 + 10 = 12 while a(2) * 10 = 2 * 10 = 20, both of which have two distinct prime factors.
MATHEMATICA
nn = 120;
c[_] := False; f[x_] := PrimeNu[x]; a[1] = j = 1; c[1] = True; u = 2;
Do[k = u; While[Or[c[k], f[j + k] != f[j k]], k++];
Set[{a[n], c[k], j}, {k, True, k}];
If[k == u, While[c[u], u++]], {n, 2, nn}];
Array[a, nn] (* Michael De Vlieger, Jul 17 2023 *)
CROSSREFS
Cf. A364261 (nondistinct factors), A001221, A027748.
Sequence in context: A332701 A342158 A344544 * A102512 A344794 A196364
KEYWORD
nonn
AUTHOR
Scott R. Shannon, Jul 16 2023
STATUS
approved