A368133 -id:A368133 - cp's OEIS frontend

A362855 a(n) = n for n <= 3; for n > 3, a(n) is the least novel multiple of k, the product of all distinct prime factors of a(n-2) that do not divide a(n-1).

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, 78, 1925, 156, 2695, 234, 3465, 52, 5775

Offset: 1

Michael De Vlieger and David James Sycamore, May 06 2023

nonn

Motivated by A362631, but instead of using one prime divisor of a(n-2) which does not divide a(n-1), the product of all such primes is used to compute a(n). - David James Sycamore, May 07 2023
From Michael De Vlieger, May 27 2023: (Start)
Primes p(k) enter the sequence in order and fairly regularly through a(20543) = p(15) = 47, immediately following primorial A002110(k-1). However, a(87723) = p(17) = 59 is the next prime to appear, following a(87722) = A002110(16).
Conjecture: all primes appear eventually, but not in order. (End)
Similar to A280866, except that the denominator here is rad(a(n-1)) instead of rad(a(n-2)). Also related to A369825. - David James Sycamore, Jan 27 2024
From Michael De Vlieger, Apr 23 2024: (Start)
Conjecture: permutation of natural numbers.
Conjecture: the smallest missing number is always either prime or a powerful number.
Primes do not appear in order; a(87723) = 59 but a(91307) = 53.
Powerful numbers appear in clusters, e.g., for n roughly between 91200 and 91320.
Though it appears primorials are always followed by primes, it is logically possible but rare that primorials can be followed by a composite number. (End)

			From _Michael De Vlieger_, Apr 23 2024: (Start)
Let rad(x) = A007947(x) and let P(x) = A002110(x).
Let S = { prime p : p | a(n-2) } and let T = { prime p : p | a(n-1) }. Then k = Product_{p in S\T} p = rad(a(n-2)*a(n-1))/rad(a(n-1)).
a(3) = 3 since rad(1*2)/rad(2) = 1; a(1) = 1, a(2) = 2, therefore a(3) = 3*1.
a(4) = 4 since rad(2*3)/rad(3) = 2; a(2) = 2, thus a(4) = 2*2.
a(5) = 6 since rad(3*4)/rad(4) = 6/2 = 3; a(3) = 3, thus a(5) = 2*3.
a(91305) = 108 and a(91306) = P(17), therefore k = 1 since rad(108) | P(17). The smallest missing number is 53, therefore a(91307) = 53*1. Related sequence A368133 = b is such that it is coincident with this sequence until b(91307) = 61, since prime(18) = 61 is the smallest prime that does not divide b(91306) = P(17). (End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..701, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue. Powerful numbers are labeled in gold (if a prime power) or blue. Primorials P(i) = A002110(i) are labeled in green.
Michael De Vlieger, Log log scatterplot of a(n) for n = 1..2^20.
Michael De Vlieger, Plot of k = pi(p) | a(n) at (x, y) = (n, k), n = 1..4581, with a color function representing multiplicity where black indicates 1, red = 2, etc. The bar of color at the bottom indicates primes in red, composite prime powers in gold, composite squarefree in green, and other numbers in blue.
Michael De Vlieger, Divisibility Based Lexically Earliest Sequence with Cellular Automaton Behavior, ResearchGate, 2024.

Cf. A002110, A007947, A280866, A306237, A362631, A368133, A369825.

nn = 100; c[] := False; m[] := 1;
f[x_] := f[x] = Times @@ FactorInteger[x][[All, 1]];
Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 2]; i = 1; j = r = 2;
Do[(While[c[Set[k, # m[#]]], m[#]++]) &[r/f[j]];
  Set[{a[n], c[k], i, j, r}, {k, True, j, k, f[j*k]}], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Feb 21 2024 *)

A007947(a(n) * a(n+1)) | A007947(a(n+1) * a(n+2)). - Peter Munn, Apr 18 2024

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

A362855 a(n) = n for n <= 3; for n > 3, a(n) is the least novel multiple of k, the product of all distinct prime factors of a(n-2) that do 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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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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