OFFSET
1,2
COMMENTS
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.
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.
LINKS
Scott R. Shannon, Table of n, a(n) for n = 1..5000.
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) = 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.
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.
CROSSREFS
KEYWORD
nonn
AUTHOR
Scott R. Shannon, Jul 10 2023
STATUS
approved