cp's OEIS Frontend

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.

%I A377193 #14 Oct 20 2024 13:46:08
%S A377193 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,
%T A377193 128,31,21,25,24,37,22,35,36,41,26,33,28,45,256,43,30,47,34,39,40,243,
%U A377193 512,53,38,49,48,59,42,125,54,61,44,63,50,729,1024,67,46,51
%N 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.
%C A377193 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.
%C A377193 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.
%C A377193 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.
%C A377193 Conjectures:
%C A377193 (i) A permutation of the positive integers in which the primes appear in order.
%C A377193 (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.
%H A377193 Michael De Vlieger, <a href="/A377193/b377193.txt">Table of n, a(n) for n = 1..10000</a>
%H A377193 Michael De Vlieger, <a href="/A377193/a377193.txt">Mathematica program</a>.
%H A377193 Michael De Vlieger, <a href="/A377193/a377193.png">Log log scatterplot of log_10(a(n))</a>, 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.
%H A377193 Michael De Vlieger, <a href="/A377193/a377193_1.png">Plot p^m | a(n) at (x,y) = (n, pi(p))</a>, 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.
%e A377193 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.
%e A377193 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.
%e A377193 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.
%Y A377193 Cf. A000027, A000040, A000961, A002110, A006530, A007947, A033845, A055932, A083720, A246655, A280864.
%K A377193 nonn
%O A377193 1,2
%A A377193 _David James Sycamore_, and _Michael De Vlieger_ Oct 19 2024