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%I A376865 #29 Nov 05 2024 21:05:20
%S A376865 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 A376865 34,39,44,35,27,16,23,38,51,52,55,42,30,29,46,57,68,65,66,49,40,36,31,
%U A376865 58,69,76,85,78,77,45,32,37,62,87,92,95,102,91,88,56,50,48
%N A376865 a(1) = 1. Thereafter, if A007947(a(n-1)) is in A002110, a(n) = smallest prime not already a term (condition [A]). Otherwise, if q is the greatest prime < Gpf(a(n-1)) which does not divide a(n-1), a(n) is the least novel multiple of q,(condition [B]).
%C A376865 A sequence with 3 distinct phases, similar to A372368.
%C A376865 Define cycle c(i) to be a run of consecutive terms beginning with a prime a(n) = prime(i) resulting from condition [A], which ends when a(n) is a term in A055932.
%C A376865 Phase I consists of consecutive closed cycles c(i) that start with a(n) = prime(i) via condition [A] and end with a term in A055932. As n increases through cycle c(i), G = gpf(a(n)) strictly decreases, and g = gpf(m(q)) is small compared to G. This phase ends at n = 4318.
%C A376865 Phase II consists of closed cycles c(i) that start with a(n) = prime(j), j > i, via condition [A] and end with a term in A055932. As n increases through cycle c(i), at times, g > G and we have a rejuvenated cycle. We may see multiple condition [B] primes, as well as runs of composite a(n) for n = 99528..155219 and n = 222811..262605. The humps in scatterplot are associated with these particular runs of composite terms. Rejuvenation of a cycle has G increment m(q) each time. A "ridge" of high m(q) values builds and grows increasingly difficult to traverse to reach G = 11, where we might have a number in A055932 and close the cycle. This phase likely ends with n = 2048704.
%C A376865 Phase III consists of condition [A] prime a(2048704) = prime(742) = 5647 and terms that follow, starting cycle c(135). As n increases, there are repeated rejuvenations and regular entry of primes through condition [B]. The repeated rejuvenations increase and expand a bank of high values of m(q) across many primes q only a few dozen iterations after new primes appear. New primes mean that prime(i) increases while in order to find a(n) in A055932, we need numbers with G = 11. Therefore, the circumstance that needs to arise to close the cycle becomes harder to achieve as n increases.
%C A376865 It is unlikely that this sequence is a permutation of natural numbers.
%C A376865 A full description of the phased behavior of this sequence is given in the link.
%H A376865 Michael De Vlieger, <a href="/A376865/b376865.txt">Table of n, a(n) for n = 1..10000</a>
%H A376865 Michael De Vlieger, <a href="/A376865/a376865.png">Log log scatterplot of a(n)</a>, n = 1..2^20.
%H A376865 Michael De Vlieger, <a href="/A376865/a376865_1.png">Log log scatterplot of G(n) = gpf(a(n))</a>, n = 1..3000000, showing the confinement of G(n) to large primes for n > 2048704.
%H A376865 Michael De Vlieger, <a href="/A376865/a376865.txt">Notes on this sequence</a>.
%H A376865 Michael De Vlieger, <a href="https://doi.org/10.13140/RG.2.2.34522.56009">Tipping Point Sequence A365865</a>, ResearchGate, 2024.
%e A376865 a(1) = 1 = A002110(0), so a(2) = 2 (smallest prime not already a term).
%e A376865 a(2) = 2 = A002110(1), so a(3) = 3.
%e A376865 a(3) = 3 not in A002110 therefore a(4) = 4, the least novel multiple of 2.
%e A376865 rad(4) = 2 so a(5) = 5 the smallest prime not already a term.
%t A376865 nn = 120; c[_] := False; m[_] := 1; f[x_] := FactorInteger[x][[All, 1]];
%t A376865 Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 2]; j = 2; v = 3;
%t A376865 Do[If[Or[IntegerQ@ Log2[j], And[EvenQ[j], Union@ Differences@ PrimePi[#] == {1}]],
%t A376865      k = v; While[c[k*m[k]], m[k]++]; k *= m[k],
%t A376865      k = Last[#]; While[And[k > 1, Divisible[j, k]], k = NextPrime[k, -1]];
%t A376865      While[c[k*m[k]], m[k]++]; k *= m[k]] &[f[j]];
%t A376865   Set[{a[n], c[k], j}, {k, True, k}];
%t A376865   If[k == v, While[c[v], v = NextPrime[v] ] ], {n, 3, nn}];
%t A376865 Array[a, nn] (* _Michael De Vlieger_, Oct 09 2024 *)
%Y A376865 Cf. A000040, A000027, A002110, A007947, A055932, A376839, A372368.
%K A376865 nonn,more
%O A376865 1,2
%A A376865 _David James Sycamore_, and _Michael De Vlieger_, Oct 07 2024
%E A376865 More terms from _Michael De Vlieger_, Oct 09 2024

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