A143232 Sum of denominators of Egyptian fraction expansion of A004001(n) - n/2.

2, 0, 2, 0, 2, 1, 2, 0, 2, 1, 3, 1, 3, 1, 2, 0, 2, 1, 3, 2, 3, 2, 4, 2, 4, 2, 3, 2, 3, 1, 2, 0, 2, 1, 3, 2, 4, 2, 4, 3, 5, 3, 5, 4, 5, 4, 5, 3, 5, 4, 5, 4, 5, 3, 5, 3, 4, 2, 4, 2, 3, 1, 2, 0, 2, 1, 3, 2, 4, 3, 4, 3, 5, 4, 6, 4, 6, 5, 7, 5, 7, 6, 7, 6, 7, 5, 7, 6, 8, 6, 8, 7, 8, 7, 8, 6, 8, 7, 8, 7, 8, 6, 8, 6, 7

Offset

1,1

Comments

A004001 is the Hofstadter-Conway $10,000 Sequence and A004001(n) - n/2 is increasingly larger versions of the batrachion Blancmange function.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

H. Rich, Hofstadter-Conway $10,000 Sequence, Programming Archives, 2008.

Eric Weisstein's World of Mathematics, Hofstadter-Conway $10,000 Sequence.

Formula

a(n) = Sum of denominators of Egyptian fraction expansion of A004001(n) - n/2 .

For practical purposes, a full Egyptian fraction algorithm is not necessary. Since the elements of A004001(n) - n/2 are either whole or their fractional part is .5, the sequence can be effected by a(n) = sefd(A004001(n) - n/2) with sefd(x) = x + 3 * (x - floor(x)) .

Example

a(43) = 5 because A004001(43) = 25, so (A004001(43) - (43/2)) = 3.5 and the Egyptian fraction expansion of 3.5 is (1/1)+(1/1)+(1/1)+(1/2), so the denominators are 1,1,1,2 which sums to 5.

Mathematica

HC[n_]:= HC[n]= If[n<3, 1, HC[HC[n-1]] +HC[n-HC[n-1]]]; (*HC=A004001*)
f[x_]:= 4*x -3*Floor[x];
A143232[n_]:= f[HC[n] -n/2];
Table[A143232[n], {n, 100}] (* G. C. Greubel, Sep 10 2024 *)

Prog

(J) a004001 =: ((] +&:$: -) $:@:<:)@.(2&<) M.
a143232 =: (+ 3 * 1&|)@:(a004001 - -:)"0
(Magma)
A004001:=[n le 2 select 1 else Self(Self(n-1))+ Self(n-Self(n-1)):n in [1..125]];
f:= func< x | 4*x - 3*Floor(x) >;
A143232:= func< n | f(A004001[n] - n/2) >;
[A143232(n): n in [1..100]]; // G. C. Greubel, Sep 10 2024
(SageMath)
@CachedFunction
def b(n): # b = A004001
    if n<3: return 1
    else: return b(b(n-1)) + b(n-b(n-1))
def f(x): return x + 3 * (x - floor(x))
def A143232(n): return f(b(n) - n/2)
[A143232(n) for n in range(1, 101)] # G. C. Greubel, Sep 10 2024

Crossrefs

Cf. A004001.

Sequence in context: A305575 A247977 A359239 * A329981 A096030 A025815

Adjacent sequences: A143229 A143230 A143231 * A143233 A143234 A143235

Keyword

nonn,easy

Author

Dan Bron (j (at) bron.us), Jul 31 2008

Status

approved