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A350150 a(1)=1; thereafter a(n+1) is the smallest unused number k such that d(k) and d(a(n)) are coprime, where d is the divisor counting function A000005.

%I A350150 #27 Dec 25 2024 00:52:12
%S A350150 1,2,4,3,9,5,16,6,25,7,36,8,49,10,64,11,81,12,625,13,100,14,121,15,
%T A350150 144,17,169,19,196,21,225,22,256,23,289,24,324,26,361,27,400,29,441,
%U A350150 30,484,31,529,33,576,34,676,35,729,18,1024,20,1296,28,2401,32,4096
%N A350150 a(1)=1; thereafter a(n+1) is the smallest unused number k such that d(k) and d(a(n)) are coprime, where d is the divisor counting function A000005.
%C A350150 A permutation of the positive integers. Identical to A137442 until a(18), with some terms in common thereafter. Numbers with the same number of divisors appear in their natural order, e.g. primes; d(p)=2, odd squarefree semiprimes; d(p*q) = 4, etc.
%C A350150 From _Michael De Vlieger_, Dec 16 2021: (Start)
%C A350150 a(2n+1) is square and d(a(2n+1)) odd. Let a(2n+1) constitute an "alpha" ray in scatterplot.
%C A350150 d(a(2n)) is even. Let a(2n) constitute a "beta" ray in scatterplot.
%C A350150 The occasion of d(a(2n)) = 6 induces a "flare" phase in the sequence, evident in scatterplot. The following term a(2n+1) is forced to have d(a(2n+1)) congruent to 1 or 5 (mod 6).
%C A350150 There are 5 flare-phases in the scatterplot associated with the occasion of d(a(2n)) = 6:
%C A350150 (I) a(18) = 12, a(19) = 625. The latter term interrupts what had been thereto and thereafter a series of square a(2n+1).
%C A350150 (II) a(54..61);
%C A350150 (III) 144..169 where a(k) with k in {152, 158, 164, 166} have d(a(k)) =/= 6, a characteristic common to subsequent phases;
%C A350150 (IV) 686..849;
%C A350150 (V) 11664..15515. (End)
%H A350150 Michael De Vlieger, <a href="/A350150/b350150.txt">Table of n, a(n) for n = 1..10000</a>
%H A350150 Michael De Vlieger, <a href="/A350150/a350150.png">Log-log scatterplot of a(n)</a> for n = 1..10^5 showing maxima in red and minima in blue. The golden line indicates the smallest missing number. We label terms that begin and end a flare phase among squares in alpha that comprise many records, and same in beta that comprise nearly all local minima.
%H A350150 Michael De Vlieger, <a href="/A350150/a350150_1.png">Log-log scatterplot of a(n)</a> for n = 1..512 annotating a(n) in black above and d(a(n)) in red below the point. The plot illustrates bifurcation of the sequence, with a(2n) such that d(a(2n)) is even in trajectory beta below, while square a(2n+1) such that d(a(2n+1)) is odd in trajectory alpha above. Terms a(2n) such that d(a(2n)) = 6 appear in blue, while a(2n+1) such that d(a(2n)) = 6 appear in red. The faint green line below represents d(a(n)) for reference.
%e A350150 a(2) = 2 because d(1) = 1, d(2) = 2 and gcd(1,2) = 1.
%e A350150 a(3) cannot be 3 since d(2) = d(3) = 2, but gcd(d(2),d(4)) = gcd(2,3) = 1, so a(3) = 4.
%t A350150 a[1] = 1; a[n_] := a[n] = Module[{k = 2, s = Array[a, n - 1], d = DivisorSigma[0, a[n - 1]]}, While[MemberQ[s, k] || ! CoprimeQ[d, DivisorSigma[0, k]], k++]; k]; Array[a, 100] (* _Amiram Eldar_, Dec 16 2021 *)
%Y A350150 Cf. A000005, A000027, A000040, A000290, A030515, A046388, A048691, A137442.
%K A350150 nonn
%O A350150 1,2
%A A350150 _David James Sycamore_, Dec 16 2021
%E A350150 More terms from _Amiram Eldar_, Dec 16 2021