

A273317


Irregular table read by rows: T(0,0) = 2 and T(n,2k) = T(n1,k)+1, T(n,2k+1) = T(n1,k)*(T(n1,k)+1) for 0 <= k < 2^(n1).


2



2, 3, 6, 4, 12, 7, 42, 5, 20, 13, 156, 8, 56, 43, 1806, 6, 30, 21, 420, 14, 182, 157, 24492, 9, 72, 57, 3192, 44, 1892, 1807, 3263442, 7, 42, 31, 930, 22, 462, 421, 176820, 15, 210, 183, 33306, 158, 24806, 24493, 599882556, 10, 90, 73, 5256, 58, 3306, 3193, 10192056
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OFFSET

0,1


COMMENTS

The first entry in row n is n+2.
The second entry in row n (n>0) is the A002378(n+2).
No number appears twice in the same row, so row n has 2^n distinct terms.
Row n and row n+1 have no elements in common.
There are infinitely many mutually disjoint rows; this fact can be used to show that the harmonic series diverges since the sum of reciprocals of entries in every row equals 1/2. This also allows a proof that every positive rational number is the sum of a finite number of distinct Egyptian fractions.
Let S(0) = {2} and for n>=1 define S(n) = {a  a = c+1 or a = c*(c+1) for some c in S(n1)}; then row n contains the elements of S(n).


LINKS

Table of n, a(n) for n=0..54.
Steven J. Kifowit, More Proofs of Divergence of the Harmonic Series.
J. C. Owings, Jr., Another Proof of the Egyptian Fraction Theorem, Amer. Math. Monthly, 75(7) (1968), 777778.


FORMULA

T(0,0) = 2, and T(n,2k) = T(n1,k)+1, T(n,2k+1) = T(n1,k)*(T(n1,k)+1) for 0 <= k < 2^(n1).
Sum_{a in row(n)} 1/a = 1/2 for all n.


EXAMPLE

The table begins:
2,
3, 6,
4, 12, 7, 42,
5, 20, 13, 156, 8, 56, 43, 1806,
6, 30, 21, 420, 14, 182, 157, 24492, 9, 72, 57, 3192, 44, 1892, 1807, 3263442,


MAPLE

A273317 := proc(n, j)
if n = 0 then
2 ;
elif type(j, 'even') then
1+procname(n1, j/2) ;
else
procname(n1, floor(j/2)) ;
%*(%+1) ;
end if;
end proc: # R. J. Mathar, May 20 2016


PROG

(Sage)
def T(n, j):
if n==0:
return 2
if j%2==0:
return T(n1, floor(j/2))+1
else:
t=T(n1, floor(j/2))
return t*(t+1)
S=[[T(n, k) for k in [0..2^n1]] for n in [0..10]]
[x for sublist in S for x in sublist]


CROSSREFS

Cf. A002378, A002061.
Sequence in context: A302848 A046202 A225642 * A328443 A122866 A097275
Adjacent sequences: A273314 A273315 A273316 * A273318 A273319 A273320


KEYWORD

nonn,tabf


AUTHOR

Tom Edgar, May 19 2016


STATUS

approved



