

A343835


Irregular table T(n, k), n > 0, k = 1..A069010(n), read by rows; the nth row contains the shortest partition of n whose values belong to A023758 and can be added without carriers in binary, in descending order.


3



1, 2, 3, 4, 4, 1, 6, 7, 8, 8, 1, 8, 2, 8, 3, 12, 12, 1, 14, 15, 16, 16, 1, 16, 2, 16, 3, 16, 4, 16, 4, 1, 16, 6, 16, 7, 24, 24, 1, 24, 2, 24, 3, 28, 28, 1, 30, 31, 32, 32, 1, 32, 2, 32, 3, 32, 4, 32, 4, 1, 32, 6, 32, 7, 32, 8, 32, 8, 1, 32, 8, 2, 32, 8, 3
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OFFSET

1,2


COMMENTS

In other words, the nth row gives the numerical values of the runs of 1's in the binary expansion of n.


LINKS



FORMULA

Sum_{k = 1..A069010(n)} T(n, k) = n.


EXAMPLE

Table begins:
1: [1]
2: [2]
3: [3]
4: [4]
5: [4, 1]
6: [6]
7: [7]
8: [8]
9: [8, 1]
10: [8, 2]
11: [8, 3]
12: [12]
13: [12, 1]
14: [14]
15: [15]
Table begins in binary:
1: [1]
10: [10]
11: [11]
100: [100]
101: [100, 1]
110: [110]
111: [111]
1000: [1000]
1001: [1000, 1]
1010: [1000, 10]
1011: [1000, 11]
1100: [1100]
1101: [1100, 1]
1110: [1110]
1111: [1111]


PROG

(PARI) row(n) = { my (rr=[]); while (n, my (z=valuation(n, 2), o=valuation(n/2^z+1, 2), r=(2^o1)*2^z); n=r; rr = concat(r, rr); ); rr }


CROSSREFS



KEYWORD

nonn,base,tabf,easy


AUTHOR



STATUS

approved



