A364137 - OEIS

%I #16 Jul 18 2023 17:16:19

%S 1,2,1,1,2,1,1,2,2,4,2,3,2,2,2,6,1,1,2,1,1,4,1,1,2,2,1,1,2,1,1,1,1,4,

%T 1,2,3,1,3,2,1,1,1,3,2,3,1,1,1,3,1,1,1,1,1,2,1,1,4,2,2,3,1,3,1,1,1,1,

%U 3,1,1,9,1,1,1,6,1,1,1,1,1,1,4,1,2,3,1,1,1,1,2,2,6,3,1,1,1,6,1

%N a(1) = 1; for n > 1, a(n) is the smallest positive number such that the sum of all terms a(1) + ... + a(n) has the same number of distinct prime factors as the product of all terms a(1) * ... * a(n).

%C The terms magnitudes show different regimes, ever increasing in average size, as a new prime factor appears in the product of all terms. In the first 5000 terms an increase in the total number of distinct prime factors of this product occurs at n = 2, 12, 127, 465, 801, 1230, 2798. After a(2798) = 1020 the sum of all terms is 881790 = 2 * 3 * 5 * 7 * 13 * 17 * 19 which contains seven distinct prime factors, while the product of all terms is 31155...000 (a number containing 5264 digits) that equals 2^4398 * 3^2902 * 5^1607 * 7^980 * 11^312 * 13^249 * 17, which also contains seven distinct prime factors. See the graph of the terms.

%C In the first 5000 terms the smallest numbers not to have appeared are 11,13,17,19,23,29,31,33,34. It is unknown if all numbers eventually appear.

%H Scott R. Shannon, <a href="/A364137/b364137.txt">Table of n, a(n) for n = 1..5000</a>.

%e 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.

%e a(3) = 1 as a(1) + a(2) + 1 = 1 + 2 + 1 = 4 while a(1) * a(2) * 1 = 1 * 2 * 1 = 2, both of which have one distinct prime factor.

%e a(12) = 3 as a(1) + ... a(11) + 3 = 1 + ... + 2 + 3 = 22 while a(1) * ... a(11) * 3 = 1 * ... * 2 * 3 = 192, both of which have two distinct prime factors.

%Y Cf. A364138 (distinct terms), A001221, A027748, A364262.

%K nonn

%O 1,2

%A _Scott R. Shannon_, Jul 10 2023