A363444
a(n) = n for n <= 3; for n > 3, a(n) is the smallest positive number that has not yet appeared that includes as factors the distinct prime factors of a(n-2) and a(n-1) that are not shared between a(n-2) and a(n-1).
Original entry on oeis.org
1, 2, 3, 6, 4, 9, 12, 8, 15, 30, 10, 18, 45, 20, 24, 60, 5, 36, 90, 25, 42, 210, 35, 48, 420, 70, 21, 120, 140, 63, 150, 280, 84, 75, 350, 126, 105, 40, 168, 315, 50, 252, 525, 80, 294, 630, 55, 462, 840, 110, 231, 1050, 220, 693, 1260, 330, 77, 1470, 660, 154, 735, 990, 308, 945, 1320, 616, 1155
Offset: 1
a(4) = 6 as a(2) = 2 and a(3) = 3 contain the distinct prime factors 2 and 3 respectively, both of which only appear in one term. Therefore a(4) is the smallest unused number that contains both 2 and 3 as factors, which is 6.
a(6) = 9 as a(4) = 6 = 2*3 and a(5) = 4 = 2*2, so 3 is the only prime factor that is not shared between these terms. Therefore a(6) is the smallest unused number that contains 3 as a factor, which is 9.
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
- Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
- Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^10, showing primes in red, composite prime powers in gold, squarefree composites in green, and other numbers in blue.
- Michael De Vlieger, Plot p(i)^e(i) | a(n) at (x,y) = (n,i) for n = 1..2048, 12X vertical exaggeration, with a color function representing e(i), where black indicates e(i) = 1, red indicates e(i) = 2, yellow-green = 3, green = 4, and blue = 5. The bar at bottom indicates primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue.
-
nn = 120; c[] := False; m[] := 1; Array[Set[{a[#], c[#]}, {#, True}] &, 3];
i = {a[2]}; j = {a[3]}; Do[q = Times @@ SymmetricDifference[i, j]; While[c[Set[k, q m[q]]], m[q]++]; Set[{a[n], c[k], i, j}, {k, True, j, FactorInteger[k][[All, 1]]}], {n, 4, nn}]; Array[a, nn] (* Michael De Vlieger, Jun 05 2023 *)
A365117
a(1) = 1. Thereafter a(n) is the least novel multiple m of the smallest prime which does not divide a(n-1) and such that m is coprime to a(n-1).
Original entry on oeis.org
1, 2, 3, 4, 9, 8, 15, 14, 27, 10, 21, 16, 33, 20, 39, 22, 45, 26, 51, 28, 57, 32, 63, 34, 69, 38, 75, 44, 81, 40, 87, 46, 93, 50, 99, 52, 105, 58, 111, 56, 117, 62, 123, 64, 129, 68, 135, 74, 141, 70, 153, 76, 147, 80, 159, 82, 165, 86, 171, 88, 177, 92, 183, 94
Offset: 1
a(2) = 2 because 2 is the smallest prime which does not divide 1, and 2 is prime to 1.
3(3) = 3 because 3 is the smallest prime which does not divide 2, and 3 is prime to 2.
a(4) = 4 since it is the second multiple of 2, the smallest prime that does not divide 3, and 4 is prime to 3.
a(5) = 9 since it is the least novel multiple of 3, the smallest prime that does not divide 4, and is the least such number prime to 4.
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
- Michael De Vlieger, Log log scatterplot of a(n), n = 1..256, showing a(n) mod 6 = 3 in red, a(n) mod 6 = 2 in large dark blue, and a(n) mod 6 = 4 in green dots. Demonstrates 2 trajectories, an early trajectory pertaining to 3 (mod 6) and a late trajectory pertaining to +-2 (mod 6).
- Michael De Vlieger, 32 X 32 table of a(n), n = 1..1024, listed in rows, with a color code according to residue (mod 6). Blue = 2 (mod 6), red = 3 (mod 6), light green = 4 (mod 6), black = 0, 1, or 5 mod 6.
- Michael De Vlieger, 30 X 30 table of a(n), n = 1..900, listed in rows, with a color code as follows: black = empty product, red = prime, gold = composite prime power, light green = squarefree semiprime, green = squarefree number with more than 2 prime factors, purple = powerful number (in A001694) that is not a prime power, blue = other numbers that are neither squarefree nor prime powers.
- Michael De Vlieger, 210 X 210 table of a(n), n = 1..44100, listed in rows with color code as immediately above, demonstrating the aggregate pattern demonstrated in the sequence. Certain residues (mod 210) seem to be devoid of squarefree numbers, perhaps since they are a product 9*m.
-
nn = 12; m[] := 1; a[1] = j = 1; m[1] = 2; c[1] = True; c[] := False; f[x_] := Block[{q}, q = 2; While[! CoprimeQ[q, x], q = NextPrime[q]]; q]; Do[{k = m[#]; While[Or[! CoprimeQ[j, k], c[# k]], k++]; If[k == m[#], While[c[m[#] #], m[#]++]]; Set[{a[n], j, c[k #]}, {k #, k #, True}]} &@ f[j], {n, 2, nn}]; Array[a, nn] (* Michael De Vlieger, Aug 22 2023 *)
-
first(n) = {my(res = vector(n)); sofar = Set([1..4]); for(i = 1, 4, res[i] = i); for(i = 5, n, res[i] = nxt(res[i-1])); res}
nxt(n) = {my(start, step); if(n % 2 == 0, start = 3; step = 6, start = 2; step = [2,4]); forstep(i = start, oo, step, s = Set(i); if(gcd(i, n) == 1 && #setminus(s, sofar) == 1, sofar = setunion(sofar, s); return(i)))} \\ David A. Corneth, Aug 22 2023
A367082
a(1), a(2) = 2. Thereafter a(n) is the least novel multiple of the greatest prime which divides precisely one of a(n-1), a(n-2) but not the other. If no such prime exists a(n) is the least novel multiple of the smallest prime dividing neither a(n-1) nor a(n-2).
Original entry on oeis.org
1, 2, 4, 3, 6, 8, 9, 12, 10, 5, 14, 7, 16, 21, 28, 15, 35, 42, 20, 49, 56, 18, 63, 70, 25, 77, 11, 84, 22, 33, 24, 44, 55, 30, 66, 88, 27, 99, 110, 40, 121, 132, 36, 143, 13, 154, 26, 39, 45, 52, 65, 50, 78, 91, 98, 104, 117, 48, 130, 156, 60, 169, 182, 105, 195, 208, 75, 221, 17, 234, 34, 51, 54, 68, 85, 80, 102, 119
Offset: 1
a(1,2) = 1,2 so a(3) = 4, the least novel multiple of 2 (which divides 2 but not 1). Since rad(2) = rad(4) = 2 there is no prime which divides one of a(2), a(3) but not the other so by the second condition of the definition a(4) = 3, the least novel multiple of the smallest prime (3) which divides neither a(2) = 2 nor a(3) = 4.
The sequence can be presented as an irregular table where row(k) starts with A008578(k), and with the exception of rows 1 and 2, ends with the earliest multiple of A008578(k+1).
The table starts:
1;
2,4;
3,6,8,9,12,10;
5,14;
7,16,21,28,15,35,42,20,49,56,18,63,70,25,77;
11,84,22,33,24,44,55,30,66,88,27,110,40,121,132,36,143;
13,154.....
T(3) is a medium trajectory, includes 3^2 but not 3*5 = 15, which appears later, in T(7). T(5) is a short trajectory, stopped by 14, does not include 25 which is delayed until T(7); T(7) is the first full trajectory, including 49, and ending with 7*11 = 77. In full and medium trajectories T(p) we see pairs of consecutive multiples of p separated by a multiple of a smaller prime. If T(prime(k)) is full it contains (prime(k+1) - 1) multiples of prime(k) and (prime(k+1) - 1)/2 multiples of smaller primes, thus T(7) contains 15 terms; see Formula.
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
- Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
- Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^12, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue.
-
nn = 120;
c[] := False; m[] := 1;
f[x_] := f[x] = FactorInteger[x][[All, 1]];
g[x_] :=
Block[{q = 2},
If[OddQ[x], q, While[Divisible[x, q], q = NextPrime[q] ]; q] ];
Array[Set[{a[#], c[#]}, {#, True}] &, 2];
i = a[1]; j = a[2]; u = 3; ri = {}; rj = {j};
Do[Set[r, Times @@ Union[ri, rj]]; If[ri == rj,
While[c[Set[k, # m[#] ] ], m[#]++] &[g[r]],
While[c[Set[k, # m[#] ] ], m[#]++] &[FactorInteger[r][[-1, 1]] ] ];
Set[{a[n], c[k], i, j, ri, rj}, {k, True, j, k, rj, f[k]}], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Nov 06 2023 *)
A379548
a(1) = 1. For n > 1, a(n) is the earliest novel squarefree number divisible by the smallest prime which does not divide a(n-1).
Original entry on oeis.org
1, 2, 3, 6, 5, 10, 15, 14, 21, 22, 30, 7, 26, 33, 34, 39, 38, 42, 35, 46, 51, 58, 57, 62, 66, 55, 70, 69, 74, 78, 65, 82, 87, 86, 93, 94, 102, 85, 106, 105, 110, 111, 114, 95, 118, 123, 122, 129, 130, 138, 115, 134, 141, 142, 159, 146, 165, 154, 174, 145, 158
Offset: 1
a(1)=1 implies a(2)=2, the smallest novel squarefree number divisible by 2, the least non divisor prime of 1. Likewise a(3)=3. a(4) must be the smallest squarefree multiple of 2, the least non divisor prime of 3, so a(4) cannot be 4, it must be 6 which is squarefree.
-
nn = 120; c[] := False; s = {1}; m[] := 1; j = 1; c[1] = True;
f[x_] := Block[{q = 2}, While[Divisible[x, q], q = NextPrime[q]]; q];
{j}~Join~Reap[Monitor[Do[k = f[j];
While[Or[c[k*m[k]], ! SquareFreeQ[k*m[k]]], m[k]++];
k *= m[k]; j = Sow[k]; c[k] = True, {n, 2, nn}], n] ][[-1, 1]] (* Michael De Vlieger, Dec 28 2024 *)
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