OFFSET
1,2
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.
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.
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn
AUTHOR
David James Sycamore, and Michael De Vlieger Oct 19 2024
STATUS
approved