Kantke's problem
N. J. A. Sloane
njas at research.att.com
Wed May 26 18:04:16 CEST 1999
A correspondent, Thomas Kantke (bytes.more at ibm.net) from Germany,
has sent a very interesting letter that is concerned with the following
way to give a "value" to a number.
Start with n and try to reach 2 by repeatedly either dividing by d where d <= the
square root or by adding or subtracting 1. The division steps are free,
but adding or subtracting 1 costs 1 point. The "value" of n (A047988) is the
smallest cost to reach 2.
He has a characterization of the Nullwertzahlen (A047836),
and some conjectures about the rate of growth of the Nullwertzahlen,
and the rate of growth of the numbers of value 1, 2, 3, etc,
but no proofs.
He would be interested to hear from anyone who can
prove anything about these numbers.
Here are some associated sequences:
%I A047836
%S A047836 2,4,8,12,16,24,32,36,40,48,56,60,64,72,80,84,96,108,112,120,128,132,
%T A047836 144
%N A047836 "Nullwertzahlen" (or "inverse prime numbers"): n=p1*p2*p3*p4*p5*...*pk, where pi are prime with p1 <= p2 <= p3 <= p4 ...; then p1 = 2 and p1*p2*...*pi >= p(i+1) for all i < k.
%C A047836 Start with n and reach 2 by repeatedly either dividing by d where d <= the square root or by adding or subtracting 1. The division steps are free, but adding or subtracting 1 costs 1 point. The "value" of n (A047988) is the smallest cost to reach 2. Sequence gives numbers with value 0.
%D A047836 Thomas Kantke, Mathematische Unterhaltungen, Spectrum der Wissenschaft, April 1993, pp. 11-13.
%F A047836 The number of a(n) <= x is conjectured to be about C * x / ln (x) , where C = 0.61...
%O A047836 1,1
%K A047836 nonn,nice,easy,more
%A A047836 Thomas Kantke (bytes.more at ibm.net)
%Y A047836 Cf. A047984-A047988.
%e A047836 From 24 we divide by 3, 2, then 2, reaching 2.
%I A047984
%S A047984 3,5,6,7,9,10,11,13,14,15,17,18,20,21,22,23,25,26,27,28,30,31,33,34,35,
%T A047984 37,39,41,42,44,45,46,47,49,50,51,52,54
%N A047984 Define value of a number m (A047988) as follows: start with m and reach 2 by repeatedly either dividing by d where d <= the square root or by adding or subtracting 1. The division steps are free, but adding or subtracting 1 costs 1 point. The "value" of m is the smallest cost to reach 2. Sequence gives numbers with value 1.
%D A047984 Thomas Kantke, Mathematische Unterhaltungen, Spectrum der Wissenschaft, April 1993, pp. 11-13.
%O A047984 1,1
%K A047984 nonn
%A A047984 Thomas Kantke (bytes.more at ibm.net)
%Y A047984 Cf. A047836, A047985-A047988.
%e A047984 7 has value 1 since we add 1 to get 8, then divide by 2 twice.
%I A047985
%S A047985 19,29,38,43,53,58,67,76,86,87,89,101,103,106,116,134,137,139,149,151,152,157,
%T A047985 163,172,174,178,197,202,203
%N A047985 Define value of a number m (A047988) as follows: start with m and reach 2 by repeatedly either dividing by d where d <= the square root or by adding or subtracting 1. The division steps are free, but adding or subtracting 1 costs 1 point. The "value" of m is the smallest cost to reach 2. Sequence gives numbers with value 2.
%D A047985 Thomas Kantke, Mathematische Unterhaltungen, Spectrum der Wissenschaft, April 1993, pp. 11-13.
%O A047985 1,1
%K A047985 nonn
%A A047985 Thomas Kantke (bytes.more at ibm.net)
%Y A047985 Cf. A047836, A047984, A047986-A047988.
%e A047985 19 has value 2 since by adding 1 we reach 20 which has value 1; there is no cheaper way to reach 2.
%I A047986
%S A047986 173,347,823,907,1237,1697,2137,2333,2423,2473,2767,2777,3253,3413,3547,3559,3623,
%T A047986 3767,4243,4273,4457
%N A047986 Define value of a number m (A047988) as follows: start with m and reach 2 by repeatedly either dividing by d where d <= the square root or by adding or subtracting 1. The division steps are free, but adding or subtracting 1 costs 1 point. The "value" of m is the smallest cost to reach 2. Sequence gives numbers with value 3.
%D A047986 Thomas Kantke, Mathematische Unterhaltungen, Spectrum der Wissenschaft, April 1993, pp. 11-13.
%O A047986 1,1
%K A047986 nonn
%A A047986 Thomas Kantke (bytes.more at ibm.net)
%Y A047986 Cf. A047836, A047985-A047988.
%I A047987
%S A047987 3976733,8053483,9942523,10197427,15126557,28623773,32269037
%N A047987 Define value of a number m (A047988) as follows: start with m and reach 2 by repeatedly either dividing by d where d <= the square root or by adding or subtracting 1. The division steps are free, but adding or subtracting 1 costs 1 point. The "value" of m is the smallest cost to reach 2. Sequence gives numbers with value 4.
%D A047987 Thomas Kantke, Mathematische Unterhaltungen, Spectrum der Wissenschaft, April 1993, pp. 11-13.
%O A047987 1,1
%K A047987 nonn
%A A047987 Thomas Kantke (bytes.more at ibm.net)
%Y A047987 Cf. A047836, A047984-A047988.
%I A047988
%S A047988 0,1,0,1,1,1,0,1,1,1,0,1,1,1,0,1,1,2,1,1,1,1,0,1,1,1,1,2,1,1,0,1,1,1,
%T A047988 0,1,2,1,0,1,1,2,1,1,1,1,0,1,1,1,1,2,1,1,0
%N A047988 a(n) = value of n, defined as follows: start with n and reach 2 by repeatedly either dividing by d where d <= the square root or by adding or subtracting 1. The division steps are free, but adding or subtracting 1 costs 1 point. The "value" of n is the smallest cost to reach 2.
%D A047988 Thomas Kantke, Mathematische Unterhaltungen, Spectrum der Wissenschaft, April 1993, pp. 11-13.
%O A047988 2,18
%K A047988 nonn
%A A047988 Thomas Kantke (bytes.more at ibm.net)
%Y A047988 Cf. A047836, A047984-A047987.
Neil Sloane
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