login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

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

%I #14 Jul 17 2023 08:50:40

%S 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,

%T 38,37,26,55,33,43,31,39,49,51,41,57,59,45,99,69,47,58,46,66,30,82,71,

%U 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

%N 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).

%C 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.

%H Scott R. Shannon, <a href="/A364261/b364261.txt">Table of n, a(n) for n = 1..10000</a>.

%H Scott R. Shannon, <a href="/A364261/a364261.png">Image of the first 50000 terms</a>. The green line is a(n) = n.

%e a(2) = 2 as a(1) + 2 = 1 + 2 = 3 while a(1) * 2 = 1 * 2 = 2, both of which have one prime factor.

%e a(3) = 6 as a(2) + 6 = 2 + 6 = 8 while a(2) * 6 = 2 * 6 = 12, both of which have three prime factors.

%t nn = 120;

%t c[_] := False; f[x_] := PrimeOmega[x]; a[1] = j = 1;

%t c[1] = True; u = 2;

%t Do[k = u; While[Or[c[k], f[j + k] != f[j k]], k++];

%t Set[{a[n], c[k], j}, {k, True, k}];

%t If[k == u, While[c[u], u++]], {n, 2, nn}];

%t Array[a, nn] (* _Michael De Vlieger_, Jul 17 2023 *)

%Y Cf. A364262 (distinct factors), A001222, A027746.

%K nonn

%O 1,2

%A _Scott R. Shannon_, Jul 16 2023