A362855 -id:A362855 - 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

A362631 Lexicographically earliest infinite sequence of distinct positive integers with a(n) = n for n <= 3, and for n > 3 a(n) is the least novel multiple of the greatest prime divisor of a(n-2) which does not divide a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 9, 10, 12, 15, 8, 20, 7, 25, 14, 30, 21, 35, 18, 28, 24, 42, 11, 49, 22, 56, 33, 63, 44, 70, 55, 77, 40, 66, 45, 88, 50, 99, 60, 110, 27, 121, 36, 132, 13, 143, 16, 26, 17, 39, 34, 52, 51, 65, 68, 78, 85, 91, 102, 104, 119, 117, 136, 130, 153, 156, 170, 169, 187, 182, 204, 195, 221, 75, 238, 80, 255, 32, 272, 19

Offset: 1

David James Sycamore, May 06 2023

nonn

The definition reflects that of A098550 in that it places a condition on a(n-2) which does not apply to a(n-1).
If there is no prime divisor of a(n-2) which does not divide a(n-1), then by empty product convention a(n) = u the least unused number.
Some primes (23,29,31,47,...) enter because of dividing a(n-2) but not a(n-1), whereas others (5,7,11,13,17,19,...) enter as least u; see Example.
With the exception of 16 all least u terms (up to a(2^28)) are primes, so it seems likely that a(47) = 16 is a one-off (fluke) term.
The scatterplot resembles a fine-toothed comb similar to those of A361629, A361133 and A361534, in which each "tooth" starts with a novel prime p and continues with a run of (mostly) alternate multiples of p and the greatest prime less than p until interrupted by the arrival of the next prime, and so forth.
The sequence, after a(1) = 1 can be represented as an irregular table in which the n-th row starts with prime(n), see Example.
Conjectured to be a permutation of the positive integers with the primes in order.

			a(4,5) = 4, 6 and since rad(4)|rad(6) a(6) = least u = 5.
a(11,12) = 8, 20 and since rad(8)|rad(20) a(13) = least u = 7.
a(44,45) = 132, 13 and gpd(132) = 11 does not divide 13, and since it is the 13th occurrence of p = 11, a(46) = 13*11 = 143.
a(45,46) = 13, 143 which forces a(47) = least u = 16 (see Comment).
a(90,91) = 69, 114 and 23 is the greatest prime dividing 69 which does not divide 114. Since 23 has not appeared earlier in the sequence a(92) = 23.
There is as yet no known formula for the row lengths of the table below. Whereas most rows terminate with a multiple of the prime they start with, there are exceptions, e.g., 47, 109. This behavior is open to explanation.
The table starts:
2;
3,4,6;
5,9,10,12,15,8,20;
7,25,14,30,21,35,18,28,24,42;
11,49,22,56,33,63,44,70,55,77,40,66,45,88,50,99,60,110,27,121,36,132;
13,143,16,26;
17,39,34,52,51,65,68,78,85,91,102,104,119,117,136,130,153,156,170,169,187,182,204,195,221,75,238,80,255,32,272;
19,....

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Scatterplot of a(n) 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..10000, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue. We accentuate powerful numbers that are not prime powers in light blue.

Cf. A007947, A098550, A361629, A361133, A361534, A362855.

mm = 3; nn = 10^4; c[] := False; q[] := 1;
Array[Set[{a[#], c[#]}, {#, True}] &, mm]; q[2]++; u = mm + 1;
i = a[mm - 1]; j = a[mm]; s = {1};
Do[t = FactorInteger[i][[All, 1]];
  If[SubsetQ[s, t], q[u]++; Set[k, u],
   p = SelectFirst[Reverse@ t, CoprimeQ[j, #] &];
   If[! IntegerQ[p], Set[k, u], While[c[p*q[p]], q[p]++]; k = p*q[p]] ];
  Set[{a[n], c[k], i, j, s}, {k, True, j, k, t}];
  If[k == u, While[c[u], u++]], {n, mm + 1, nn}];
Array[a, nn]

A383342 Lexicographically earliest infinite sequence of distinct positive integers such that the number following any consecutive pair x, y of terms is the smallest novel number divisible by R(x,y) = rad(x*y)/rad(gcd(x,y)).

Original entry on oeis.org

1, 2, 4, 3, 6, 8, 9, 12, 10, 15, 18, 20, 30, 21, 70, 60, 42, 35, 90, 84, 105, 40, 126, 210, 5, 168, 420, 25, 252, 630, 45, 14, 840, 75, 28, 1050, 120, 7, 1260, 150, 49, 1470, 180, 56, 315, 240, 98, 525, 270, 112, 735, 300, 140, 63, 330, 770, 147, 660, 1540, 189

Offset: 1

David James Sycamore and Michael De Vlieger, Apr 22 2025

nonn

For numbers x, y let f(x) = {prime p: p|x} and f(y) = {prime q: q|y} be the sets of distinct prime factors of x and y respectively, then R(x,y) is the product of the primes which occur once only in the union of f(x) and f(y). There are two conditions for entry of a prime following adjacent terms x, y: (i). rad(x) = rad(y) implies that the next term is the smallest number not yet seen in the sequence (which could be prime); (ii). rad(x) = A002110(k), rad(y) = A002110(k)/prime(m), m <= k implies prime(m). After computation of 2^24 terms the only primes found are 2,3,5,7. Furthermore in the same data range, if a given composite squarefree number k appears there seems no guarantee that all numbers m with rad(m) = rad(k) will also appear (a(5) = 6 and subsequently we see only 12 and 18; a(9) = 10 and thereafter we see only 20,40,50,80,100). Also the primorial numbers do not appear in order (e.g. A002110(11) appears before A002110(10)).

			It follows from the definition that the first two terms must be a(1) = 1, a(2) = 2. R(1,2) = rad(2)/rad(1) = 2 and since 2 is already a term, a(3) = 4.
Since a(2) = 2 and a(3) = 4 have the same rad it follows that a(4) = 3, the smallest novel number.
R(4,3) = rad(12)/rad(1) = 6, so a(5) = 6, since 6 has not occurred earlier.
R(3,6) = rad(18)/rad(3) = 2, so a(6) = 8, the least novel multiple of 2.
R(6,8) = rad(48)/rad(2) = 6/2 = 3 so a(7) = 9, the least novel multiple of 3.
a(23) = 126, a(24) = 210 and R(126,210) = rad(126*210)/rad(42) = 210/42 = 5, which has not occurred earlier, so a(25) = 5.

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..2^16, showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where purple indicates powerful numbers that are not prime powers. Primorials are highlighted in large bright green points.
Michael De Vlieger, Plot p^k | a(n) at (x,y) = (n, pi(p)) for n = 1..2^11, 12X vertical exaggeration, with a color function showing k = 1 in black, k = 2 in red, ... maximum value of k in reference range in magenta. The color bar under the plot indicates numbers as immediately above, red = prime, etc.

Cf. A002110, A005117, A007947, A362855, A368133, A369825.

nn = 120; rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
c[] := False; m[] := 1; i = 1; j = 2; c[1] = c[2] = True;
{1, 2}~Join~Reap[Do[k = rad[i*j]/rad[GCD[i, j]];
  While[c[k*m[k]], m[k]++]; k *= m[k];
  Set[{c[k], i, j}, {True, j, k}]; Sow[k],
{n, 3, nn}] ][[-1, 1]] (* Michael De Vlieger, Apr 25 2025 *)

A370740 a(1) = 1. Thereafter a(n) is the least novel k such that A007947(k*a(n-1)) is the smallest number in A002110 which is not already a term.

Original entry on oeis.org

Offset: 1

David James Sycamore, Feb 28 2024

nonn

In other words, for n > m, where a(m) = A002110(r), a(n) is the least novel k such that rad(k*a(n-1)) = A002110(r+1).
Sequence is same as A362855 and A368133 until a(57) = 32.
Conjectured to be a permutation of the positive integers (A000027), with primorials, primes and prime powers in natural order.

			a(1) = 1--> a(2) = 2 since 2 is the least primorial exceeding 1.
a(2) = 2--> a(3) = 3 since 2*3 = 6, the next primorial, and no k < 3 is such that rad(k*2) = 6.
a(3) = 3--> a(4) = 4 since rad(3*4) = rad(12) = 6.
a(4) = 4-->a(5) = 6 since rad(4*6) = rad(24) = 6.
a(58,59,60,61) = 2310,13,4620,26 = P(5), prime(6), 2*P(5), 2*prime(6).

Cf. A000027, A000040, A000961, A002110, A007947, A362855, A368133.

For m >= 1, a(n) = P(m) = A002110(m)-->a(n+1) = prime(m+1), a(n+2) = 2*P(m), a(n+3) = 2*prime(m+1); (see last in Example).

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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A362631 Lexicographically earliest infinite sequence of distinct positive integers with a(n) = n for n <= 3, and for n > 3 a(n) is the least novel multiple of the greatest prime divisor of a(n-2) which does not divide a(n-1).

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A383342 Lexicographically earliest infinite sequence of distinct positive integers such that the number following any consecutive pair x, y of terms is the smallest novel number divisible by R(x,y) = rad(x*y)/rad(gcd(x,y)).

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Mathematica

A370740 a(1) = 1. Thereafter a(n) is the least novel k such that A007947(k*a(n-1)) is the smallest number in A002110 which is not already a term.

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