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A286603
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Restricted growth sequence computed for sigma, A000203.
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20
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 6, 11, 12, 13, 13, 14, 10, 15, 16, 17, 18, 19, 13, 20, 14, 17, 21, 22, 23, 24, 18, 25, 26, 27, 26, 28, 29, 20, 22, 30, 17, 31, 32, 33, 34, 24, 26, 35, 36, 37, 24, 38, 27, 39, 24, 39, 40, 30, 20, 41, 42, 31, 43, 44, 33, 45, 46, 47, 31, 45, 24, 48, 49, 50, 35, 51, 31, 41, 40, 52, 53, 47, 33, 54, 55, 56, 39, 57, 30, 58, 59, 41, 60
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OFFSET
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1,2
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COMMENTS
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When filtering sequences (by equivalence class partitioning), this sequence can be used instead of A000203, because for all i, j it holds that: a(i) = a(j) <=> A000203(i) = A000203(j) <=> A286358(i) = A286358(j).
Note that the latter equivalence indicates that this is also the restricted growth sequence of A286358.
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LINKS
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EXAMPLE
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Construction: we start with a(1)=1 for sigma(1)=1 (where sigma = A000203), and then after, for all n > 1, whenever the value of sigma(n) has not been encountered before, we set a(n) to the least natural number k not already in sequence among a(1) .. a(n-1), otherwise [whenever sigma(n) = sigma(m), for some m < n], we set a(n) = a(m), i.e., to the same value that was assigned to a(m).
For n=2, sigma(2) = 3, not encountered before, thus we allot for a(2) the least so far unused number, which is 2, thus a(2) = 2.
For n=3, sigma(3) = 4, not encountered before, thus we allot for a(3) the least so far unused number, which is 3, thus a(3) = 3.
For n=4, sigma(4) = 7, not encountered before, thus we allot for a(4) the least so far unused number, which is 4, thus a(4) = 4.
For n=5, sigma(5) = 6, not encountered before, thus we allot for a(5) the least so far unused number, which is 5, thus a(5) = 5.
For n=6, sigma(6) = 12, not encountered before, thus we allot for a(6) the least so far unused number, which is 6, thus a(6) = 6.
And this continues for n=7..10 because also for those n sigma obtains fresh new values, so here a(n) = n up to n = 10.
But then comes n=11, where sigma(11) = 12, a value which was already encountered at n=6 for the first time, thus we set a(11) = a(6) = 6.
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MATHEMATICA
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With[{nn = 93}, Function[s, Table[Position[Keys@ s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Array[DivisorSigma[1, #] &, nn]] (* Michael De Vlieger, May 12 2017, Version 10 *)
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PROG
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(PARI)
rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences, invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences, invec[i], i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset, vec, bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
write_to_bfile(1, rgs_transform(vector(10000, n, A000203(n))), "b286603.txt");
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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