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Scott R. Shannon, <a href="/A364138/b364138_1.txt">Table of n, a(n) for n = 1..1000</a>
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The terms magnitudes show different regimes, even ever increasing in average size, as a new prime factor appears in the product of all terms. In the first 1000 terms an increase in the total number of distinct prime factors of this product occurs at n = 2, 3, 9, 22, 56, 159, 385, 714. After a(714) = 118404 the sum of all terms is 11741730 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 23 which contains eight distinct prime factors, while the product of all terms is 11122...000 (a number containing 2585 digits) that equals 2^1738 * 3^1136 * 5^664 * 7^486 * 11^299 * 13^237 * 23 * 29^46, which also contains eight distinct prime factors. See the graph of the terms.
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editing
editing
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In the first 1000 terms the smallest numbers not to have appeared are 5,11,13,14,17,19,23,26,29. It is unknown if all numbers eventually appear.
Scott R. Shannon, <a href="/A364138/b364138_1.txt">Table of n, a(n) for n = 1..1000</a>
a(2) = 2 as 2 has not previously appeared and a(1) + 2 = 1 + 2 = 3 while a(1) * 2 = 1 * 2 = 2, both of which have one distinct prime factor.
a(2) = 2 as 2 has not previously appeared and a(1) + 2 = 1 + 2 = 3 while a(1) * 2 = 1 * 2 = 2, both of which have one distinct prime factor.a(3) = 3 as 3 has not previously appeared and a(1) + a(2) + 3 = 1 + 2 + 3 = 6 while a(1) * a(2) * 3 = 1 * 2 * 3 = 6, both of which have two distinct prime factors.a(9) = 15 as 15 has not previously appeared and a(1) + ... a(8) + 15 = 1 + ... + 12 + 15 = 60 while a(1) * ... a(8) * 15 = 1 * ... * 12 * 15 = 1866240, both of which have three distinct prime factors.
a(9) = 15 as 15 has not previously appeared and a(1) + ... a(8) + 15 = 1 + ... + 12 + 15 = 60 while a(1) * ... a(8) * 15 = 1 * ... * 12 * 15 = 1866240, both of which have three distinct prime factors.
The terms magnitudes show different regimes, even increasing in average size, as a new prime factor appears in the product of all terms. In the first 1000 terms an increase in the total number of distinct prime factors of this product occurs at terms n = 2, 3, 9, 22, 56, 159, 385, 714. After a(714) = 118404 the sum of all terms is 11741730 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 23 which contains eight distinct prime factors, while the product of all terms is 11122...000 (a number containing 2585 digits) that equals 2^1738 * 3^1136 * 5^664 * 7^486 * 11^299 * 13^237 * 23 * 29^46, which also contains eight distinct prime factors. See the graph of the terms.
dist uniq
a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared 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).
The terms magnitudes show different regimes, even increasing in average size, as a new prime factor appears in the product of all terms. In the first 1000 terms an increase in the total number of distinct prime factors of this product occurs at terms 2, 3, 9, 22, 56, 159, 385, 714. After a(714) = 118404 the sum of all terms is 11741730 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 23 which contains eight distinct prime factors, while the product of all terms is 11122...000 (a number containing 2585 digits) that equals 2^1738 * 3^1136 * 5^664 * 7^486 * 11^299 * 13^237 * 23 * 29^46, which also contains eight distinct prime factors. See the graph of the terms.
a(2) = 2 as 2 has not previously appeared and a(1) + 2 = 1 + 2 = 3 while a(1) * 2 = 1 * 2 = 2, both of which have one distinct prime factor.a(3) = 3 as 3 has not previously appeared and a(1) + a(2) + 3 = 1 + 2 + 3 = 6 while a(1) * a(2) * 3 = 1 * 2 * 3 = 6, both of which have two distinct prime factors.a(9) = 15 as 15 has not previously appeared and a(1) + ... a(8) + 15 = 1 + ... + 12 + 15 = 60 while a(1) * ... a(8) * 15 = 1 * ... * 12 * 15 = 1866240, both of which have three distinct prime factors.