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A364138
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).
3
1, 2, 3, 4, 8, 6, 9, 12, 15, 10, 20, 24, 16, 40, 25, 27, 18, 30, 36, 48, 45, 21, 42, 84, 144, 80, 28, 60, 72, 90, 120, 50, 64, 126, 150, 108, 147, 35, 70, 105, 7, 98, 162, 180, 168, 96, 54, 100, 200, 75, 63, 32, 160, 240, 140, 220, 300, 330, 210, 630, 810, 360, 960, 264, 336, 420, 672
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 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.
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.
LINKS
EXAMPLE
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.
CROSSREFS
Cf. A364137 (nondistinct terms), A001221, A027748, A364262.
Sequence in context: A222256 A367870 A223540 * A300868 A349239 A176077
KEYWORD
nonn
AUTHOR
Scott R. Shannon, Jul 10 2023
STATUS
approved