A377193 Lexicographically earliest infinite sequence of distinct positive integers such that any term j = a(n-1) with primorial kernel is followed by a prime, whereas any other term is followed by a number with prime factors p < q = Gpf(j) which do not divide j.

1, 2, 3, 4, 5, 6, 7, 8, 11, 9, 16, 13, 10, 27, 32, 17, 12, 19, 14, 15, 64, 23, 18, 29, 20, 81, 128, 31, 21, 25, 24, 37, 22, 35, 36, 41, 26, 33, 28, 45, 256, 43, 30, 47, 34, 39, 40, 243, 512, 53, 38, 49, 48, 59, 42, 125, 54, 61, 44, 63, 50, 729, 1024, 67, 46, 51

Offset: 1

Author

David James Sycamore, and Michael De Vlieger Oct 19 2024

Keywords

• nonn

Comments

Following j = a(n-1), a term in A005932, a(n) is the smallest prime not already listed. Otherwise a(n) = smallest novel product of powers of non divisor primes of j; a number of the form: Product_{i = 0..k} p_i^e_i; p_i a prime < q = Gpf(j) which does not divide j, e_i >= 0, k = the number of primes p_i < q which do not divide j.

Adjacent terms are coprime and the greedy algorithm implied by the definition forces naked prime p to appear in advance of any multiple m*p of p; m >1.

Prime powers enter the sequence early, consequent to j having a single non divisor prime. A power of 3 is always followed by a power of 2.

Conjectures:

(i) A permutation of the positive integers in which the primes appear in order.

(ii)The sequence obeys Selcoe's theorem (see A280864) regarding numbers that have the same squarefree kernel, namely: Construct a sequence S_r = { m*r : rad(m) | r } = { k : rad(k) = r }, squarefree r. Terms w in S_r appear in this sequence in order. This is to say, for example, that for r = 6, terms in A033845 = {6, 12, 18, 24, 36, 48, 54, ...} appear in order.

Examples

			a(1) = 1 implies a(2) = 2 since A007947(1) = A002110(1) = 1, and 2 is the earliest unrecorded prime so far, and likewise a(3) = 3. Since rad(3) = 3 is not a primorial number a(4) = 2^2 = 4, the smallest novel number derived from 2, the only non divisor prime of 3 and < 3.
a(8) = 8 implies a(9) = 11 because 8 is a term in A055932. The non divisor primes of 11 and < 11 are 2,3,5,7  and the smallest number which can be composed using some or all of these primes is a(10) = 3^2 = 9 (since 2,3,4,5,6,7,8 have all occurred previously). Consequently a(11) = 2^4 = 16, the smallest novel power of 2.
a(195) = 154 = 2*7*11, the non divisor primes < 11 are 3 and 5, so a(196) = 405 = 3^4*5 since all smaller candidates (3,5,9,15,25,45,75,81,125,135,243,375) have already appeared.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Mathematica program.
• Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..25000, showing primes in red, perfect prime powers in gold, squarefree composites in green, numbers in A332785 in blue, and A286708 in purple.
• Michael De Vlieger, Plot p^m | a(n) at (x,y) = (n, pi(p)), n = 1..2048, 4X vertical exaggeration, with a color function showing exponent m = 1 in black, m = 2 in red, ..., m = 11 in magenta. The color index at bottom uses the color key described immediately above.

Crossrefs

Cf. A000027, A000040, A000961, A002110, A006530, A007947, A033845, A055932, A083720, A246655, A280864.