A368108 -id:A368108 - cp's OEIS frontend

A368133 a(1,2,3) = 1,2,3; let j = a(n-1), M(n) = Product_{i = 1..n-2} { p a distinct prime: p | a(i), gcd(p, j) = 1 }. For n > 3, a(n) is the least novel multiple of M(n) if M(n) > 1; otherwise a(n) is the least novel multiple of A053669(j), the smallest prime which does not divide j.

1, 2, 3, 4, 6, 5, 12, 10, 9, 20, 15, 8, 30, 7, 60, 14, 45, 28, 75, 42, 25, 84, 35, 18, 70, 21, 40, 63, 50, 105, 16, 210, 11, 420, 22, 315, 44, 525, 66, 140, 33, 280, 99, 350, 132, 175, 198, 245, 264, 385, 24, 770, 27, 1540, 36, 1155, 26, 2310, 13, 4620, 39, 3080

Offset: 1

David James Sycamore, Dec 13 2023

nonn

M(n) is a squarefree number whose prime factors are the distinct primes which divide a(m), m <= n-2, but do not divide j. M(n) > 1 implies there exists at least one term prior to j having a prime divisor which does not divide j, and M(n) is the product of all such primes. If, for any term a(m), m <= n-2, every prime factor of a(m) also divides j, then M(n) = 1, the empty product.
Primorial a(n-1) implies prime a(n); see Formula.
Conjectured to be a permutation of the positive integers.
Compare with A368108 which has a slightly different definition but works in a similar way.
From Michael De Vlieger, Jan 05 2024: (Start)
This sequence is the same as A362855 for 91306 terms.
A362855(91306) = a(91306) = A002110(17),
A362855(91307) = 53 = prime(16), a(91307) = 61 = prime(18),
A362855(91308) = A002110(17)/prime(16), a(91308) = 2*A002110(17).
Thereafter the sequences diverge. It seems unlikely that the 2 sequences will become coincident again as n increases beyond 91308. (End)

			a(1, 2, 3) = 1, 2, 3. M(4) = 2 because 2 | a(2) but does not divide a(3); 2 is the only a(m), m < 3, with this property, so a(4) = 4, the least novel multiple of 2.
Now we have a(1,2,3,4) = 1,2,3,4. M(5) = 3 because 3 | a(3) but does not divide a(4); 3 is the only a(m), m < 4, with this property, so a(5) = 2*3 = 6, the least novel multiple of 3.
We now have a(1..5) = 1, 2, 3, 4, 6. M(6) = 1, the empty product, because there is no prime which divides some a(m), m < 5, which does not also divide a(n-1) = 6. This situation invokes the second condition of the definition, so a(6) = 1*5, the least novel multiple of A053669(6) = 5, the smallest prime which does not divide 6. Consequently a(7) = 2*6 = 12 because no prime dividing a(1..5) also divides 5.
The same situation arises again at a(13) = 30 = 2*3*5; every prime divisor of a(m), m < 13, is 2, 3, or 5, which again invokes the second condition, M(14) = 1, the empty product, so a(14) = 1*7, since A053669(30) = 7. Consequently a(15) = 2*7 = 14.
a(91307) = 61 (whereas A362855(91307) = 53; point of divergence from A362855).

Michael De Vlieger, Log log scatterplot of a(n) and b(n), n = 1..2^20, where a(n) is shown in red, b(n) = A362855(n) is shown in blue. Black indicates where a(n) = b(n).

Cf. A002110, A053669, A351495, A362855, A368108.

nn = 10^5; c[] := False; m[] := 1;
Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 3]; j = 3;
s = {2}; r = Max[s]; c[3] = False;
q[x_] := Block[{qq = 2}, While[Divisible[x, qq], qq = NextPrime[qq]]; qq];
Do[(If[# == 1,
        Set[k, NextPrime[r]],
        While[Or[c[#], # == j] &[# m[#]], m[#]++];
          Set[k, # m[#]]] &[Times @@ Complement[s, #]];
          s = Union[s, #];
     If[Last[#] > r, r = Last[#]]) &@ FactorInteger[j][[All, 1]];
  Set[{a[n], c[j], j}, {k, True, k}], {n, 4, nn}];
Array[a, nn] (* Michael De Vlieger, Jan 05 2024 *)

When for some m, a(m) = A002110(n), a primorial number, a(m+1) = prime(n+1), a(m+2) = 2*A002110(n), and a(m+3) = 2*prime(n+1); see Example.

a(n) = A362855(n), for 1 <= n <= 91306 (see link and Example).

More terms from Michael De Vlieger, Jan 05 2024

A379746 a(1)=1. For n>1 if a(n-1)=A002110(k), a(n)=prime(k+1). Otherwise a(n) is the smallest novel number whose prime factors have already occurred as previous terms.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 7, 14, 21, 28, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 98, 100, 105, 108, 112, 120, 125, 126, 128, 135, 140, 144, 147, 150, 160, 162, 168, 175, 180

Offset: 1

David James Sycamore, Jan 01 2025

Equivalent definition: Lexicographically earliest infinite sequence of distinct positive integers such that a(n) is the smallest novel number having prime power factorization Product_p_i^e_i where p_i is the least nondivisor prime of at most e_i distinct terms a(j); 1<=j<=n-1.

A permutation of the positive integers with prime powers q^k appearing in order (k>=1), and whose underlying sequence of least nondivisor primes is a permutation of A053669. Also, for distinct x, y; x

No multiple m*p (m>1) of a prime p can occur before p itself is a term.

From Michael De Vlieger, Jan 02 2025: (Start)

Efficient method of generating the sequence:

Define row k to be a(A363061(k)+1..A363061(k+1)).

Define R(i) to be { m <= i : rad(m) | i } = tensor product of prime power factor ranges of i that do not exceed i.

Then row k contains R(A002110(k+1)) \ R(A002110(k)).

Row 0 is R(1) = {1}.

Row 1 is R(2)\R(1) = {1, 2} \ {1} = {2},

i.e., {row 2 of A162306} \ {row 1 of A162306}

= {first A363061(1) terms of A000079} \ {1}.

Row 2 is R(6)\R(2) = {1, 2, 3, 4, 6} \ {1, 2} = {3, 4, 6},

where R(6) = row 6 of A162306 = first A363061(2) terms of A003586.

Row 3 is R(30)\R(6)

= {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30} \ {1, 2, 3, 4, 6}

= {5, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30},

where R(30) = row 30 of A162306 = first A363061(3) terms of A051037, etc.

Therefore, for k > 1, within each row, terms strictly increase from prime(k) to primorial A002110(k).

Furthermore, a(1..A363061(k)) is a permutation of R(A002110(k)), hence the sequence is infinite and a permutation of natural numbers. (End)

			a(1) = 1 = A002110(0) therefore a(2) = A053669(1) = 2.
a(2) = 2 = A002110(1) therefore a(3) = A053669(2) = 3.
a(3) = 3 is not a primorial term so a(4)=4 = 2^2 is the smallest novel number whose prime factors do not exceed 3.
Using the second definition we have a(1,2,3,4)=1,2,3,4
                  with least nondivisor primes 2,3,2,3 respectively. Therefore a(5)=2^1*3^1=6, the smallest novel number whose prime factors (2,3) are nondivisor primes of the first 4 terms, and whose exponents do not exceed the number of times these primes have occurred in the underlying sequence of least nondivisor primes.

Michael De Vlieger, Table of n, a(n) for n = 1..19985 (to include A002110(8).)
Michael De Vlieger, Efficient Wolfram program to generate this sequence. (Note: pay attention to dataset size in A363061).
Michael De Vlieger, Log log scatterplot of a(n), n = 1..19985.
Michael De Vlieger, Plot p^m | a(n) at (x,y) = (n, pi(p)), n = 1..2048, 12X vertical exaggeration, with a color function representing m = 1 in black, m = 2 in red, m = 3 in orange, ..., highest value of m in the dataset in magenta.

Cf. A000027, A000040, A002110, A007947, A053669, A162306, A351495, A363061, A368108, A377566.

nn = 120; kk = 12;
c[] := False; m[] := 0; h = 0; q = j = 1; u = 2;
f[x_] := f[x] = FactorInteger[x][[All, 1]];
MapIndexed[Set[P[First[#2] - 1], #1] &, FoldList[Times, 1, Prime@ Range[kk]]];
{1}~Join~Reap[Do[
    If[j == P[h],
      If[h == kk, Break[]]; k = Prime[h + 1]; h++; q = Prime[h],
      k = u; While[Or[c[k], ! AllTrue[f[k], # <= q &]], k++]];
    j = Sow[k]; c[k] = True; If[k == u, While[c[u], u++] ],
{n, 2, nn}] ][[-1, 1]] (* Michael De Vlieger, Jan 02 2025 *)

From Michael De Vlieger, Jan 02 2025: (Start)
a(A363061(k)) = A002110(k).
a(A363061(k)+1) = prime(k).
Seen as a table T(j,k), k = 1..A363061(j)-A363061(j-1) for j > 0, row 0 = {1},
  row j = {row A002110(j) of A162306} \ {row A002110(j-1) of A162306}. (End)

Showing 1-2 of 2 results.

cp's OEIS Frontend

A368133 a(1,2,3) = 1,2,3; let j = a(n-1), M(n) = Product_{i = 1..n-2} { p a distinct prime: p | a(i), gcd(p, j) = 1 }. For n > 3, a(n) is the least novel multiple of M(n) if M(n) > 1; otherwise a(n) is the least novel multiple of A053669(j), the smallest prime which does not divide j.

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