A230303 Let M(1)=0 and for n >= 2, let B(n)=M(ceiling(n/2))+M(floor(n/2))+2, M(n)=2^B(n)+M(floor(n/2))+1; sequence gives M(n).

0, 5, 129, 4102, 87112285931760246646623899502532662132742, 1852673427797059126777135760139006525652319754650249024631321344126610074239106

Offset: 1

Author

N. J. A. Sloane, Oct 24 2013; Mar 26 2014

Keywords

• nonn

Comments

M(n) is the smallest value of k such that A228085(k) = n. For example, 129 is the first time a 3 appears in A228085 (and is therefore the first term in A230092). M(4) = 4102 is the first time a 4 appears in A228085 (and is therefore the first term in A227915).

Examples

			The terms are a(1) = 0, a(2) = 2^2+0+1, a(3) = 2^7+0+1, a(4) = 2^12+5+1, a(5) = 2^136+5+1, a(6) = 2^160+129+1, a(7) = 2^4233+129+1, a(8) = 2^8206+4102+1, a(9) = 2^k+4102+1 with k=2^136+4110, ... .
The length (in bits) of the n-th term is A230302(n)+1.
		

Links

• Max A. Alekseyev and N. J. A. Sloane, On Kaprekar's Junction Numbers, arXiv:2112.14365, 2021; Journal of Combinatorics and Number Theory 12:3 (2022), 115-155.
• Index entries for Colombian or self numbers and related sequences

Crossrefs

Cf. A228085, A230092, A227915, A230093, A230302 (for B(n)), A230864.

Smallest number m such that u + (sum of base-b digits of u) = m has exactly n solutions, for bases 2 through 10: A230303, A230640, A230638, A230867, A238840, A238841, A238842, A238843, A006064.

Programs

Maple

f:=proc(n) option remember; local B, M;
if n<=1 then RETURN([0,0]);
else
if (n mod 2) = 0 then B:=2*f(n/2)[2]+2;
else B:=f((n+1)/2)[2]+f((n-1)/2)[2]+2; fi;
M:=2^B+f(floor(n/2))[2]+1; RETURN([B,M]); fi;
end proc;
[seq(f(n)[2],n=1..6)];

Formula

Define i by 2^(i-1) < n <= 2^i. Then it appears that

a(n) = 2^2^2^...^2^x

a tower of height i+3, containing i+2 2's, where x is in the range 0 < x <= 1.

For example, if n=7, i=3, and

a(7) = 2^4233+130 = 2^2^2^2^2^.88303276...

Note also that i+2 = A230864(a(n)).

Extensions

a(1)-a(8) were found by Donovan Johnson, Oct 22 2013.