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A351578
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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).
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4
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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, 20, 21, 22, 18, 7, 16, 22, 6, 22, 18, 17, 23, 24, 25, 15, 16, 26, 16, 27, 28, 23, 29, 30, 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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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(# of trailing 0's) 1-runs (allotted #)
-----+----------------------------------------------------------------------
1 : 1 [1], 0 [1] 1
2 : 2 [10], 1 [1] 2
3 : 4 [100], 2 [1] 3
4 : 3 [11], 0 [2] 4
5 : 8 [1000], 3 [1] 5
6 : 5 [101], 0 [1,1] 6
7 : 10 [1010], 1 [1,1] 7
8 : 16 [10000], 4 [1] 8
9 : 6 [110], 1 [2] 9
10 : 9 [1001], 0 [1,1] 6
11 : 18 [10010], 1 [1,1] 7
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.
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PROG
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(PARI)
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; };
v109812 = readvec("b109812_to10e5.txt"); \\ Prepared from b-file data with gawk ' { print $2 } '
up_to = #v109812;
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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