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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).
+10
3
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, 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, 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
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.
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.
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).
+10
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
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.
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.
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).
+10
2
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, 38, 37, 26, 55, 33, 43, 31, 39, 49, 51, 41, 57, 59, 45, 99, 69, 47, 58, 46, 66, 30, 82, 71, 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
COMMENTS
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.
EXAMPLE
a(2) = 2 as a(1) + 2 = 1 + 2 = 3 while a(1) * 2 = 1 * 2 = 2, both of which have one prime factor.
a(3) = 6 as a(2) + 6 = 2 + 6 = 8 while a(2) * 6 = 2 * 6 = 12, both of which have three prime factors.
MATHEMATICA
nn = 120;
c[_] := False; f[x_] := PrimeOmega[x]; a[1] = j = 1;
c[1] = True; u = 2;
Do[k = u; While[Or[c[k], f[j + k] != f[j k]], k++];
Set[{a[n], c[k], j}, {k, True, k}];
If[k == u, While[c[u], u++]], {n, 2, nn}];
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