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%I A351578 #18 Apr 07 2022 12:14:19
%S A351578 1,2,3,4,5,6,7,8,9,6,7,10,6,11,12,13,14,6,15,16,15,16,17,18,7,16,19,
%T A351578 20,21,22,18,7,16,22,6,22,18,17,23,24,25,15,16,26,16,27,28,23,29,30,
%U A351578 23,22,6,31,16,7,32,33,15,33,18,22,18,27,33,16,22,34,23,17,25,27,16,35,36,37,38,32,28,32,39,18,40,16
%N A351578 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(f(i)) = A007814(f(j)) and A278222(f(i)) = A278222(f(j)), for all i, j >= 1, where f(k) = A109812(k).
%C A351578 Restricted growth sequence transform of the ordered pair [A007814(A109812(n)), A046523(A005940(1+A109812(n)))].
%C A351578 The sequence allots a new distinct number for each newly encountered combination of the 2-adic valuation of A109812 (A351964), and the multiset of the lengths of 1-runs in the odd part of A109812 (A351965). See the examples.
%C A351578 For all i, j: a(i) = a(j) => A352889(i) = A352889(j).
%H A351578 Antti Karttunen, <a href="/A351578/b351578.txt">Table of n, a(n) for n = 1..100000</a>
%H A351578 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%e A351578    n   A109812(n)  [base-2]   A351964(n)           Lengths of       a(n)
%e A351578                               (# of trailing 0's)  1-runs       (allotted #)
%e A351578 -----+----------------------------------------------------------------------
%e A351578    1 :          1       [1],  0                    [1]               1
%e A351578    2 :          2      [10],  1                    [1]               2
%e A351578    3 :          4     [100],  2                    [1]               3
%e A351578    4 :          3      [11],  0                    [2]               4
%e A351578    5 :          8    [1000],  3                    [1]               5
%e A351578    6 :          5     [101],  0                    [1,1]             6
%e A351578    7 :         10    [1010],  1                    [1,1]             7
%e A351578    8 :         16   [10000],  4                    [1]               8
%e A351578    9 :          6     [110],  1                    [2]               9
%e A351578   10 :          9    [1001],  0                    [1,1]             6
%e A351578   11 :         18   [10010],  1                    [1,1]             7
%e A351578 Because the combinations of the multiset of 1-runs in the binary expansion of A109812(n) and the number of trailing zeros in it (A351964) are unique for n = 1 .. 9, a unique increasing number (starting from 1) is allotted for each, and a(n) = n for n <= 9. On the other hand, at n=10, the binary expansion is [1001], for which these two measures are equal to that of binary expansion [101] found first time at n=6, therefore the rgs-transform allots for 10 the same number as for 6, and a(10) = a(6) = 6. At n=11, the binary expansion is [10010], where these two measures coincide with that of [1010] found first time at n=7, therefore a(10) = a(7) = 7.
%o A351578 (PARI)
%o A351578 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 A351578 v109812 = readvec("b109812_to10e5.txt"); \\ Prepared from b-file data with gawk ' { print $2 } '
%o A351578 up_to = #v109812;
%o A351578 A109812(n) = v109812[n];
%o A351578 A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
%o A351578 A007814(n) = valuation(n,2);
%o A351578 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
%o A351578 v351578 = rgs_transform(vector(up_to, n, [A007814(A109812(n)), A046523(A005940(1+A109812(n)))]));
%o A351578 A351578(n) = v351578[n];
%Y A351578 Cf. A007814, A109812, A278222 (A286622), A351963, A351964, A351965, A352888, A352889.
%K A351578 nonn,base
%O A351578 1,2
%A A351578 _Antti Karttunen_, Apr 07 2022

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