The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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). 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 (list; graph; refs; listen; history; text; internal format)
 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(n-1)}; then row n contains the elements of S(n). LINKS 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), 777-778. FORMULA 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). 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(n-1, j/2) ;     else         procname(n-1, 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(n-1, floor(j/2))+1     else:         t=T(n-1, floor(j/2))         return t*(t+1) S=[[T(n, k) for k in [0..2^n-1]] 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 22 12:56 EDT 2021. Contains 345380 sequences. (Running on oeis4.)