A368133 - OEIS

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

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OFFSET

1,2

COMMENTS

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)

LINKS

Table of n, a(n) for n=1..62.

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).

FORMULA

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).

EXAMPLE

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).

MATHEMATICA

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]];