This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A372368 #14 May 25 2024 15:42:21
%S A372368 1,2,3,4,5,6,7,10,9,8,11,14,15,12,13,22,21,20,18,17,26,33,28,25,24,19,
%T A372368 34,39,44,35,30,23,38,51,52,55,42,40,27,16,29,46,57,68,65,66,49,45,36,
%U A372368 31,58,69,76,85,78,77,56,50,48,37,62,87,92,95,102,91,88,63
%N A372368 a(1) = 1. Thereafter, if A007947(a(n-1)) is a term in A002110, a(n) is the smallest prime not already in the sequence. Otherwise a(n) is the smallest novel multiple of p, the greatest prime < g = gpd(a(n-1)).
%C A372368 In other words, if the squarefree kernel (radical) of a(n-1) is a primorial number, a(n) is the smallest prime which is not already a term. Otherwise a(n) is the smallest novel multiple of the greatest prime p < g = gpd(a(n-1)). Initially, the arrival of a prime term p > 2 produces a run of multiples (m(q)) of primes q < p with q decrementing as n increases, until reaching a term whose kernel is a primorial number, whereupon the next prime comes in and the process is repeated.
%C A372368 A term whose radical is in A002110 occurs only if the multiplier m (of prime q) is in A002110, either with q|m, or with q the smallest prime > gpd(m). Thus all multiples of every prime will appear and it is therefore conjectured that this sequence is a permutation of the positive integers, A000027.
%C A372368 The only route to a power of 2 in the sequence is if no prime q < p, and its associated multiplier m(q) in the descending order of multiples provoked by a higher prime produces a primorial kernel. In such cases a power of 2 is only possible if preceded by a power of 3. Powers of 2 occur later and later as the sequence extends.
%C A372368 Whereas it seems obvious that a prime can only appear consequent to a primorial kernel, this is not always the case (see Example). Despite this it appears that primes arriving from the second part of the definition do not disturb the natural order of primes in the sequence (a prime from the second part of the definition is the one expected by the first).
%C A372368 See notes link for basic description of tipping point behavior in this sequence between n = 114767..171742, associated with gpf(m) > p, conspicuous in scatterplot. The sequence seems to be permanently transformed after n = 3406825.
%H A372368 Michael De Vlieger, <a href="/A372368/b372368.txt">Table of n, a(n) for n = 1..10000</a>
%H A372368 Michael De Vlieger, <a href="/A372368/a372368.png">Log log scatterplot of a(n)</a>, n = 1..3600000.
%H A372368 Michael De Vlieger, <a href="/A372368/a372368_1.png">Log log scatterplot of a(n)</a>, n = 1..120000, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where purple shows powerful numbers that are not prime powers.
%H A372368 Michael De Vlieger, <a href="/A372368/a372368.txt">Notes on this sequence</a>.
%e A372368 rad(a(1)) = rad(1) = 1 = A002110(0), hence a(2) = 2, the smallest novel prime.
%e A372368 rad(a(2)) = rad(2) = 2 = A002110(1), so a(3) = 3, the smallest novel prime.
%e A372368 a(3) = 3 is not a term in A002110 so a(4) = 4, the least novel multiple of p = 2.
%e A372368 rad(a(4)) = rad(4) = 2 = A002110(1), so a(5) = next novel prime = 5.
%e A372368 Define condition [A] to be the entry of smallest missing prime given primorial rad(a(n-1)), and define condition [B] to be the entry of a(n) = m * p.
%e A372368 The following illustrates the cycle following a(32) = prime(9) = 23. This cycle is "full", ending with a power of 2. We mark condition [A] in the last column, all unmarked terms come about through condition [B].
%e A372368 prime factors
%e A372368 n a(n) 2 3 5 7 11 13 17 19 23 29 m g
%e A372368 -----------------------------------------------
%e A372368 32 23 . . . . . . . . 1 (1) 23 [A]
%e A372368 33 38 1 . . . . . . 1 2 19
%e A372368 34 51 . 1 . . . . 1 3 17
%e A372368 35 52 2 . . . . 1 4 13
%e A372368 36 55 . . 1 . 1 5 11
%e A372368 37 42 1 1 . 1 6 7
%e A372368 38 40 3 . 1 8 5
%e A372368 39 27 . 3 9 3
%e A372368 40 16 4 8 2
%e A372368 41 29 . . . . . . . . . 1 (1) 29 [A]
%e A372368 a(41) = 29 since rad(a(40)) = rad(16) = 2 = A002110(1).Earliest example of a prime that comes in through condition [B]:
%e A372368 prime factors
%e A372368 n a(n) 2 3 5 7 11 13 17 19 ~ 313 317 331 m g
%e A372368 ------------------------------------------------------------
%e A372368 3473 2850 1 1 2 . . . . 1 150 19
%e A372368 3474 2805 . 1 1 . 1 . 1 165 17
%e A372368 3475 2730 1 1 1 1 . 1 210 13
%e A372368 3476 2695 . . 1 2 1 245 11
%e A372368 3477 2317 . . . 1 . . . . ~ . . 1 331 7
%e A372368 3478 317 . . . . . . . . ~ . 1 1 317
%e A372368 3479 1252 2 . . . . . . . ~ 1 4 313
%e A372368 a(3496) = 2695 = 5*7^2*11, which is not in A002110, so a(3497) = smallest novel multiple of 7 = m*7 for m = 331 (prime). Thus a(3497) = 7*331 = 2317. This implies a(3498) = m*317 for m = 1 (since 317 is the greatest prime < 331). Condition [B] becomes the most frequent source of primes in the sequence as n increases.
%t A372368 nn = 1000; c[_] := False; m[_] := 1; P = FoldList[Times, 1, Prime@ Range[120]];
%t A372368 a[1] = j = 1; c[1] = True; v = 2;
%t A372368 Monitor[Do[
%t A372368 If[MemberQ[P, Times @@ #[[All, 1]]],
%t A372368 k = v,
%t A372368 While[c[Set[k, # m[#]]], m[#]++] &[
%t A372368 Prime[PrimePi[#[[-1, 1]] ] - 1] ] ] &[FactorInteger[j]];
%t A372368 Set[{a[n], c[k], j}, {k, True, k}];
%t A372368 If[k == v, While[Or[c[v], CompositeQ[v]], v++]], {n, 2, nn}], n];
%t A372368 Array[a, nn]
%Y A372368 Cf. A002110, A007947, A055932.
%K A372368 nonn
%O A372368 1,2
%A A372368 _David James Sycamore_ and _Michael De Vlieger_ Apr 28 2024