A379746 - OEIS

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.

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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<y, if rad(x)=rad(y) then x appears before y.

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)

LINKS

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.

FORMULA

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)

EXAMPLE

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.

MATHEMATICA

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 *)

CROSSREFS