cp's OEIS Frontend

A273317 Irregular table read by rows: T(0,0) = 2 and T(n,2k) = T(n-1,k)+1, T(n,2k+1) = T(n-1,k)*(T(n-1,k)+1) for 0 <= k < 2^(n-1).

%I A273317 #21 May 20 2016 13:54:24
%S A273317 2,3,6,4,12,7,42,5,20,13,156,8,56,43,1806,6,30,21,420,14,182,157,
%T A273317 24492,9,72,57,3192,44,1892,1807,3263442,7,42,31,930,22,462,421,
%U A273317 176820,15,210,183,33306,158,24806,24493,599882556,10,90,73,5256,58,3306,3193,10192056
%N A273317 Irregular table read by rows: T(0,0) = 2 and T(n,2k) = T(n-1,k)+1, T(n,2k+1) = T(n-1,k)*(T(n-1,k)+1) for 0 <= k < 2^(n-1).
%C A273317 The first entry in row n is n+2.
%C A273317 The second entry in row n (n>0) is the A002378(n+2).
%C A273317 No number appears twice in the same row, so row n has 2^n distinct terms.
%C A273317 Row n and row n+1 have no elements in common.
%C A273317 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.
%C A273317 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(n-1)}; then row n contains the elements of S(n).
%H A273317 Steven J. Kifowit, <a href="http://stevekifowit.com/pubs/harm2.pdf">More Proofs of Divergence of the Harmonic Series</a>.
%H A273317 J. C. Owings, Jr., <a href="http://www.jstor.org/stable/2315211">Another Proof of the Egyptian Fraction Theorem</a>, Amer. Math. Monthly, 75(7) (1968), 777-778.
%F A273317 T(0,0) = 2, and T(n,2k) = T(n-1,k)+1, T(n,2k+1) = T(n-1,k)*(T(n-1,k)+1) for 0 <= k < 2^(n-1).
%F A273317 Sum_{a in row(n)} 1/a = 1/2 for all n.
%e A273317 The table begins:
%e A273317 2,
%e A273317 3, 6,
%e A273317 4, 12, 7, 42,
%e A273317 5, 20, 13, 156, 8, 56, 43, 1806,
%e A273317 6, 30, 21, 420, 14, 182, 157, 24492, 9, 72, 57, 3192, 44, 1892, 1807, 3263442,
%p A273317 A273317 := proc(n,j)
%p A273317     if n = 0 then
%p A273317         2 ;
%p A273317     elif type(j,'even') then
%p A273317         1+procname(n-1,j/2) ;
%p A273317     else
%p A273317         procname(n-1,floor(j/2)) ;
%p A273317         %*(%+1) ;
%p A273317     end if;
%p A273317 end proc: # _R. J. Mathar_, May 20 2016
%o A273317 (Sage)
%o A273317 def T(n,j):
%o A273317     if n==0:
%o A273317         return 2
%o A273317     if j%2==0:
%o A273317         return T(n-1,floor(j/2))+1
%o A273317     else:
%o A273317         t=T(n-1,floor(j/2))
%o A273317         return t*(t+1)
%o A273317 S=[[T(n,k) for k in [0..2^n-1]] for n in [0..10]]
%o A273317 [x for sublist in S for x in sublist]
%Y A273317 Cf. A002378, A002061.
%K A273317 nonn,tabf
%O A273317 0,1
%A A273317 _Tom Edgar_, May 19 2016