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A322026 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(i) = A007814(j) and A007949(i) = A007949(j), for all i, j, where A007814 and A007949 give the 2- and 3-adic valuations of n.

%I A322026 #41 Sep 08 2024 13:51:34
%S A322026 1,2,3,4,1,5,1,6,7,2,1,8,1,2,3,9,1,10,1,4,3,2,1,11,1,2,12,4,1,5,1,13,
%T A322026 3,2,1,14,1,2,3,6,1,5,1,4,7,2,1,15,1,2,3,4,1,16,1,6,3,2,1,8,1,2,7,17,
%U A322026 1,5,1,4,3,2,1,18,1,2,3,4,1,5,1,9,19,2,1,8,1,2,3,6,1,10,1,4,3,2,1,20,1,2,7,4,1,5,1,6,3
%N A322026 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(i) = A007814(j) and A007949(i) = A007949(j), for all i, j, where A007814 and A007949 give the 2- and 3-adic valuations of n.
%C A322026 Restricted growth sequence transform of the ordered pair [A007814(n), A007949(n)].
%C A322026 For all i, j:
%C A322026   A305900(i) = A305900(j) => a(i) = a(j),
%C A322026   a(i) = a(j) => A122841(i) = A122841(j),
%C A322026   a(i) = a(j) => A244417(i) = A244417(j),
%C A322026   a(i) = a(j) => A322316(i) = A322316(j) => A072078(i) = A072078(j).
%C A322026 If and only if a(k) > a(i) for all k > i then k is in A003586, - _David A. Corneth_, Dec 03 2018
%C A322026 That is, A003586 gives the positions of records (1, 2, 3, 4, 5, ...) in this sequence.
%C A322026 Sequence A126760 (without its initial zero) and this sequence are ordinal transforms of each other.
%H A322026 Antti Karttunen, <a href="/A322026/b322026.txt">Table of n, a(n) for n = 1..65537</a>
%H A322026 Jon Maiga, <a href="http://sequencedb.net/s/A322026">Computer-generated formulas for A322026</a>, Sequence Machine.
%F A322026 For s = A003586(n), a(s) = n = a((6k+1)*s) = a((6k-1)*s), where s is the n-th 3-smooth number and k > 0. - _David A. Corneth_, Dec 03 2018
%F A322026 A065331(n) = A003586(a(n)). - _David A. Corneth_, Dec 04 2018
%F A322026 From _Antti Karttunen_, Sep 08 2024: (Start)
%F A322026 a(n) = Sum{k=1..n} [A126760(k)==A126760(n)], where [ ] is the Iverson bracket.
%F A322026 a(n) = A071521(A065331(n)). [Found by Sequence Machine and also by LODA miner]
%F A322026 a(n) = A323884(25*n). [Conjectured by Sequence Machine]
%F A322026 (End)
%o A322026 (PARI)
%o A322026 up_to = 65537;
%o A322026 rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
%o A322026 A007814(n) = valuation(n,2);
%o A322026 A007949(n) = valuation(n,3);
%o A322026 v322026 = rgs_transform(vector(up_to, n, [A007814(n), A007949(n)]));
%o A322026 A322026(n) = v322026[n];
%o A322026 (PARI)
%o A322026 A065331(n) = (3^valuation(n, 3)<<valuation(n, 2)); \\ From A065331
%o A322026 A071521(n) = { my(t=1/3); sum(k=0, logint(n, 3), t*=3; logint(n\t, 2)+1); }; \\ From A071521.
%o A322026 A322026(n) = A071521(A065331(n)); \\ _Antti Karttunen_, Sep 08 2024
%Y A322026 Cf. A003586 (positions of records, the first occurrence of n), A007814, A007949, A065331, A071521, A072078, A087465, A122841, A126760 (ordinal transform), A322316, A323883, A323884.
%Y A322026 Cf. also A247714 and A255975.
%K A322026 nonn
%O A322026 1,2
%A A322026 _Antti Karttunen_, Dec 03 2018