login
A364261
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 prime factors as a(n-1) * a(n).
2
1, 2, 6, 10, 14, 13, 15, 3, 7, 19, 9, 11, 23, 4, 28, 53, 5, 17, 25, 35, 21, 27, 29, 34, 22, 38, 37, 26, 55, 33, 43, 31, 39, 49, 51, 41, 57, 59, 45, 99, 69, 47, 58, 46, 66, 30, 82, 71, 77, 61, 20, 44, 52, 12, 148, 68, 60, 196, 92, 36, 220, 212, 103, 62, 18, 78, 122, 73, 8, 127, 67, 79, 74, 97, 85
OFFSET
1,2
COMMENTS
The terms are concentrated along a line just above a(n) = n, resulting in twenty-four fixed points in the first 50000 terms. These begin 1, 2, 21, 116, 141, 292, 477, 700, ... . See the linked image. In the same range the smallest unseen number is 342, 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 prime factor.
a(3) = 6 as a(2) + 6 = 2 + 6 = 8 while a(2) * 6 = 2 * 6 = 12, both of which have three prime factors.
MATHEMATICA
nn = 120;
c[_] := False; f[x_] := PrimeOmega[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. A364262 (distinct factors), A001222, A027746.
Sequence in context: A376652 A074929 A095270 * A263827 A281974 A080324
KEYWORD
nonn
AUTHOR
Scott R. Shannon, Jul 16 2023
STATUS
approved