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

Author

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

Keywords

• nonn
• easy

Comments

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

Examples

			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.
		

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.

Crossrefs

Cf. A004001.

Programs

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

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 *)

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

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)) .