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A356867 For n >= 1, write n = 3^m + k, where m >= 0 is the greatest power of 3 <= n, and k is in the range 0 <= k < 3^(m+1) - 3^m, then for n such that k=0, a(n)=n, and for n such that k > 0, a(n) is the smallest prime multiple p*a(k), p != 3, that is not already a term.

%I A356867 #64 Jul 01 2025 10:08:03
%S A356867 1,2,3,5,4,6,10,8,9,7,14,15,25,20,12,50,16,18,35,28,30,125,40,24,100,
%T A356867 32,27,11,22,21,55,44,42,70,56,45,49,98,75,175,140,60,250,80,36,245,
%U A356867 196,150,625,200,48,500,64,54,77,110,105,275,88,84,350,112,90,343
%N A356867 For n >= 1, write n = 3^m + k, where m >= 0 is the greatest power of 3 <= n, and k is in the range 0 <= k < 3^(m+1) - 3^m, then for n such that k=0, a(n)=n, and for n such that k > 0, a(n) is the smallest prime multiple p*a(k), p != 3, that is not already a term.
%C A356867 Any prime p may be used to generate a sequence D(p) of this kind. The present sequence is D(3), and D(2) is the Doudna sequence, A005940.
%C A356867 Conjectured to be a permutation of the positive integers in which the primes appear in order.
%C A356867 From _Antti Karttunen_, Sep 16 2023: (Start)
%C A356867 The conjecture is true: Sequence is a permutation of natural numbers. By definition it is injective, and the surjectivity is guaranteed by the fact that there are infinitely many such n > k encountered by the greedy algorithm that a(n) will be a multiple of a(k), and "the smallest prime multiple" condition guarantees that all multiples of a(k) will eventually appear. That the primes and A100484 appear in order follows from the formulas a(3^m + 1) = prime(m+2), and a(3^m + 2) = 2*prime(m+2).
%C A356867 If the base-3 representation of n-1 has the base-3 representation of k-1 as its suffix, then a(n) is a multiple of a(k). For example, A007089(16-1) = 120, and A007089(43-1) = 1120, thus the former is the suffix of the latter, and a(16) = 50 indeed divides a(43) = 250.
%C A356867 (End)
%H A356867 Michael De Vlieger, <a href="/A356867/b356867.txt">Table of n, a(n) for n = 1..19683</a> (19683 = 3^9)
%H A356867 Michael De Vlieger, <a href="/A356867/a356867.png">Fan style ternary tree</a> showing a(n) for n = 1..3^9, with a heat map color function for level m where 3^m is blue, smaller values are bluer, and larger are yellow-green. The smallest value in level m is shown in purple and largest is shown in red.
%H A356867 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A356867 a(3^m + 1) = prime(m+2) for m >= 1.
%F A356867 Conjectures from _Jianing Song_, Nov 23 2022: (Start)
%F A356867 (1) a(3^m+2) = 2*prime(m+2) for m >= 2. - [The conjecture is true because a(2) = 2 and 3^m + 2 < 3^(1+m) + (3^m) + 1 for all m - _Antti Karttunen_, Sep 16 2023]
%F A356867 (2) For n > m >= 1, a(3^n+3^m+1) = prime(m+2)^2 for n = m+1; prime(n+2)*prime(m+2)^2 for n >= m+2.
%F A356867 (3) For n > m >= 1, a(3^n+3^m+2) = 4*prime(n+2) for n >= 3, m = 1; 2*prime(m+2)^2 for n = m+1, m >= 2; 2*prime(m+2)*prime(m+3) for n = m+2, m >= 2; 2*prime(n+2)*prime(m+2)^2 for n >= m+3, m >= 2. (End)
%F A356867 From _Antti Karttunen_, Sep 17 2023: (Start)
%F A356867 If A053735(n) = 1, then a(n) = n, otherwise a(n) = A365424(n) * a(A365459(n)).
%F A356867 For all n >= 1, A007949(a(n)) = A007949(n) and a(3*n) = 3*a(n).
%F A356867 For n >= 1, a(3^n - 1) = 2^(2n - 1), a(A048473(n)) = 2^(2*(n-1)).
%F A356867 These are conjectures so far:
%F A356867 For n >= 1, a(3^n - 2) = 10^(n-1).
%F A356867 For n >= 2, a(3^n - 3) = A002023(n-2) = 6*4^(n-2).
%F A356867 (End)
%e A356867 n=1=3^0+0 so a(1)=1. n=2=3^0+1 so k=1 and a(2)=2. Similarly a(3)=3 and a(9)=9.
%e A356867 n=10=3^2+1, therefore k=1 and a(1)=1 so a(10)=1*7=7 (since 2 and 5 have already occurred).
%t A356867 Block[{a, c, i, j, k, m, t, nn}, nn = 64; m = 1; i = 2; p = Prime[i]; c[_] = False; Do[Set[{m, k}, {1, n - p^Floor[Log[p, n]]}]; If[k == 0, Set[{a[n], c[n]}, {n, True}], While[Set[t, Prime[m] a[k]]; Or[m == i, c[t]], m++]; Set[{a[n], c[t]}, {t, True}]], {n, nn}]; Array[a, nn] ] (* _Michael De Vlieger_, Sep 01 2022 *)
%o A356867 (Python)
%o A356867 from sympy import nextprime
%o A356867 from sympy.ntheory import digits
%o A356867 from itertools import count, islice
%o A356867 def b(n): return n - 3**(len(digits(n,3)) - 2)
%o A356867 def agen():
%o A356867     aset, alst = set(), [None]
%o A356867     for n in count(1):
%o A356867         k = b(n)
%o A356867         if k == 0: an = n
%o A356867         else:
%o A356867             ak, p = alst[k], 2
%o A356867             while p == 3 or p*ak in aset: p = nextprime(p)
%o A356867             an = p*ak
%o A356867         yield an; aset.add(an); alst.append(an)
%o A356867 print(list(islice(agen(), 64))) # _Michael S. Branicky_, Sep 02 2022
%o A356867 (PARI)
%o A356867 up_to = 19683;
%o A356867 A356867list(up_to) = { my(v=vector(up_to),met=Map(),h=0,ak); for(i=1,#v,if(1==vecsum(digits(i,3)), v[i] = i; h = i, ak = v[i-h]; forprime(p=2,,if(3!=p && !mapisdefined(met,p*ak), v[i] = p*ak; break))); mapput(met,v[i],i)); (v); };
%o A356867 v356867 = A356867list(up_to);
%o A356867 A356867(n) = v356867[n]; \\ _Antti Karttunen_, Sep 15 2023
%Y A356867 Cf. A007089, A007949, A011655, A048473, A100484, A053735, A364958 (fixed points), A365390 (inverse permutation), A365424, A365459, A365462 [= a(n)-n], A365463 [= gcd(a(n),n)], A365464, A365465, A365717 [= A348717(a(1+n))], A365719 [= A046523(a(1+n))], A365721 [= omega(a(1+n))], A365722 [= bigomega(a(1+n))].
%Y A356867 Cf. also A005940, A364611, A364628 for variants D(2), D(5) and D(7).
%K A356867 nonn,look
%O A356867 1,2
%A A356867 _David James Sycamore_, Sep 01 2022
%E A356867 More terms from _Michael De Vlieger_, Sep 01 2022