

A100002


Start with a sequence of 1's, then replace every other 1 with a 2; then replace every third of the remaining 1's with a 3 and every third of the remaining 2's with a 3; then replace every fourth remaining 1, 2 or 3 with a 4; and so on. The limiting sequence is shown here.


6



1, 2, 1, 2, 3, 3, 1, 2, 4, 4, 3, 4, 1, 2, 5, 5, 3, 5, 1, 2, 4, 5, 3, 4, 6, 6, 1, 2, 6, 3, 7, 7, 6, 4, 7, 7, 5, 6, 1, 2, 5, 3, 8, 8, 7, 4, 8, 8, 1, 2, 6, 7, 3, 6, 5, 8, 4, 8, 5, 6, 9, 9, 1, 2, 9, 3, 10, 10, 9, 4, 10, 10, 7, 8, 9, 5, 7, 10, 1, 2, 9, 7, 3, 4, 9, 6, 11, 11, 10, 11
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OFFSET

1,2


COMMENTS

The position of the 1's is given by A000960.  T. D. Noe, Oct 26 2004


LINKS

T. D. Noe, Table of n, a(n) for n=1..10000
T. D. Noe, Plot of first 5000 terms
A post on sci.math.research newsgroup.


FORMULA

a(1, j)=1 for all j>=1; a(n, j)=a(n1, j) except when #{i<=j s.t. a(n1, i)=a(n1, j)} is multiple of n, in which case a(n, j)=n; a(j) is the limit of the (stationary) a(n, j) when n tends to infinity.
It appears that the maximal value among the first n terms grows like sqrt(4n/3).
Note that the first occurrence of n is bounded by A000960; that is, A100287(n) <= A000960(n), with equality only for n=1.  T. D. Noe, Nov 12 2004


EXAMPLE

Here are the first 6 stages in the construction:
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1...
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2...
1 2 1 2 3 3 1 2 1 2 3 3 1 2 1 2 3 3 1 2 1 2 3 3 1 2 1 2 3 3...
1 2 1 2 3 3 1 2 4 4 3 4 1 2 1 2 3 3 1 2 4 4 3 4 1 2 1 2 3 3...
1 2 1 2 3 3 1 2 4 4 3 4 1 2 5 5 3 5 1 2 4 5 3 4 1 2 1 2 3 3...
1 2 1 2 3 3 1 2 4 4 3 4 1 2 5 5 3 5 1 2 4 5 3 4 6 6 1 2 6 3...
...


MATHEMATICA

nn=100; t=Table[1, {nn}]; done=False; k=1; While[ !done, k++; cnt=Table[0, {k1}]; Do[If[t[[i]]<k, cnt[[t[[i]]]]++; If[Mod[cnt[[t[[i]]]], k]==0, t[[i]]=k]], {i, nn}]; done=(Max[cnt]<k)]; t (* T. D. Noe *)
a[n_] := Fold[Function[{b1, b2}, Fold[Function[{a1, a2}, ReplacePart[a1, Pick[Position[a1, a2], Take[Flatten[Array[{Array[0 &, b2  1], 1} &, Length[a1]]], Length[Position[a1, a2]]], 1] > b2]], b1, Range[b2  1]]], Array[1 &, n], Range[2, 2 Sqrt[n/Pi] + 1]]; a[100] (* Birkas Gyorgy, Feb 06 2011 *)


PROG

/* C */ #define MAXVAL 2048 /* Large enough... */ unsigned int counts[MAXVAL][MAXVAL]; /* Initialized at all 0 */ unsigned int seq_value (void) /* Successive calls return values in the sequence, in order. */ { unsigned int value; unsigned int i; value = 1; for ( i=2; i<MAXVAL; i++ ) if ( ++counts[i][value] >= i ) { counts[i][value] = 0; value = i; } return value; }


CROSSREFS

Cf. A100287 (first occurrence of n).
Sequence in context: A071766 A007305 A112531 * A227909 A210805 A057041
Adjacent sequences: A099999 A100000 A100001 * A100003 A100004 A100005


KEYWORD

easy,nice,nonn


AUTHOR

David A. Madore, Oct 25 2004


STATUS

approved



