Re: [math-fun] Freeman Dyson integer problem
210526315789473684 appears twice in the OEIS, but only once in a relevant entry: %I A094676 %S A094676 210526315789473684,3103448275862068965517241379,410256, %T A094676 510204081632653061224489795918367346938775, %U A094676 6101694915254237288135593220338983050847457627118644067796,7101449275362318840579,8101265822784,91011235955056179775280898876404494382022471 %N A094676 Least n-transposable number. %C A094676 A k-transposable number,2<=k<=9, is one equal to k times the number whose digits are merely a cyclic permutation of its own leftmost digit to the right. %D A094676 H. Camous, Jouer Avec Les Maths, "Chassez le naturel", Section I, Problem 3 pp. 20;31-2, Les Editions D'Organisation, Paris 1984. %D A094676 L. A. Graham, Ingenious Mathematical Problems and Methods, "End At The Beginning", Problem 72 pp. 44;212-3 Dover NY 1959. %F A094676 n prepended to n*(10^m - n)/(10*n - 1), where m=A094224(n)-1. %e A094676 a(4)=410256=4*102564 %K A094676 fini,nonn %O A094676 2,1 %A A094676 Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 07 2004 - though the definition doesn't look right! I will edit it. Anyway, could someone send me more terms of the sequence 210526315789473684, ... of numbers with the Freeman Dyson doubling property? Neil
We need to be a little more precise about what property we are looking for. What I thought we were saying is: start with N, remove the leading digit A, and consider 10N + A. N is good if 10N + A is exactly half of N. If that summary is correct, then 105263157894736842 is also good, because it's exactly twice 52631578947368421. Yours in lawyering about leading zeroes... On Fri, Mar 27, 2009 at 12:37 AM, N. J. A. Sloane <njas@research.att.com>wrote:
210526315789473684 appears twice in the OEIS, but only once in a relevant entry:
%I A094676 %S A094676 210526315789473684,3103448275862068965517241379,410256, %T A094676 510204081632653061224489795918367346938775, %U A094676 6101694915254237288135593220338983050847457627118644067796,7101449275362318840579,8101265822784,91011235955056179775280898876404494382022471 %N A094676 Least n-transposable number. %C A094676 A k-transposable number,2<=k<=9, is one equal to k times the number whose digits are merely a cyclic permutation of its own leftmost digit to the right. %D A094676 H. Camous, Jouer Avec Les Maths, "Chassez le naturel", Section I, Problem 3 pp. 20;31-2, Les Editions D'Organisation, Paris 1984. %D A094676 L. A. Graham, Ingenious Mathematical Problems and Methods, "End At The Beginning", Problem 72 pp. 44;212-3 Dover NY 1959. %F A094676 n prepended to n*(10^m - n)/(10*n - 1), where m=A094224(n)-1. %e A094676 a(4)=410256=4*102564 %K A094676 fini,nonn %O A094676 2,1 %A A094676 Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 07 2004
- though the definition doesn't look right! I will edit it.
Anyway, could someone send me more terms of the sequence 210526315789473684, ... of numbers with the Freeman Dyson doubling property?
Neil
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If we go from the original problem statement (from the original message in the thread), then you have it backwards (and leading zeroes are therefore prohibited). The original question clearly states that we are moving a digit from the end of a number to the front. The numbers with the Freeman-Dyson doubling property are exactly the repeating blocks of the numbers 2/19, 3/19, 4/19, 5/19, 6/19, 7/19, 8/19, and 9/19, and those numbers formed by taking one of these repeating blocks and repeating it one or more times. On Fri, Mar 27, 2009 at 1:17 PM, Allan Wechsler <acwacw@gmail.com> wrote:
We need to be a little more precise about what property we are looking for. What I thought we were saying is: start with N, remove the leading digit A, and consider 10N + A. N is good if 10N + A is exactly half of N.
If that summary is correct, then 105263157894736842 is also good, because it's exactly twice 52631578947368421.
Yours in lawyering about leading zeroes...
On Fri, Mar 27, 2009 at 12:37 AM, N. J. A. Sloane <njas@research.att.com
wrote:
210526315789473684 appears twice in the OEIS, but only once in a relevant entry:
%I A094676 %S A094676 210526315789473684,3103448275862068965517241379,410256, %T A094676 510204081632653061224489795918367346938775, %U A094676
6101694915254237288135593220338983050847457627118644067796,7101449275362318840579,8101265822784,91011235955056179775280898876404494382022471
%N A094676 Least n-transposable number. %C A094676 A k-transposable number,2<=k<=9, is one equal to k times the number whose digits are merely a cyclic permutation of its own leftmost digit to the right. %D A094676 H. Camous, Jouer Avec Les Maths, "Chassez le naturel", Section I, Problem 3 pp. 20;31-2, Les Editions D'Organisation, Paris 1984. %D A094676 L. A. Graham, Ingenious Mathematical Problems and Methods, "End At The Beginning", Problem 72 pp. 44;212-3 Dover NY 1959. %F A094676 n prepended to n*(10^m - n)/(10*n - 1), where m=A094224(n)-1. %e A094676 a(4)=410256=4*102564 %K A094676 fini,nonn %O A094676 2,1 %A A094676 Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 07 2004
- though the definition doesn't look right! I will edit it.
Anyway, could someone send me more terms of the sequence 210526315789473684, ... of numbers with the Freeman Dyson doubling property?
Neil
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_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Indeed; I should have gone back and refreshed my memory. But it turns out that even if you permit my generalization, this only has the effect of adding 1/19 to the list below; not a very interesting or liberating extension. On Fri, Mar 27, 2009 at 4:25 PM, Dave Blackston <hyperdex@gmail.com> wrote:
If we go from the original problem statement (from the original message in the thread), then you have it backwards (and leading zeroes are therefore prohibited). The original question clearly states that we are moving a digit from the end of a number to the front.
The numbers with the Freeman-Dyson doubling property are exactly the repeating blocks of the numbers 2/19, 3/19, 4/19, 5/19, 6/19, 7/19, 8/19, and 9/19, and those numbers formed by taking one of these repeating blocks and repeating it one or more times.
On Fri, Mar 27, 2009 at 1:17 PM, Allan Wechsler <acwacw@gmail.com> wrote:
We need to be a little more precise about what property we are looking for. What I thought we were saying is: start with N, remove the leading digit A, and consider 10N + A. N is good if 10N + A is exactly half of N.
If that summary is correct, then 105263157894736842 is also good, because it's exactly twice 52631578947368421.
Yours in lawyering about leading zeroes...
On Fri, Mar 27, 2009 at 12:37 AM, N. J. A. Sloane <njas@research.att.com
wrote:
210526315789473684 appears twice in the OEIS, but only once in a relevant entry:
%I A094676 %S A094676 210526315789473684,3103448275862068965517241379,410256, %T A094676 510204081632653061224489795918367346938775, %U A094676
6101694915254237288135593220338983050847457627118644067796,7101449275362318840579,8101265822784,91011235955056179775280898876404494382022471
%N A094676 Least n-transposable number. %C A094676 A k-transposable number,2<=k<=9, is one equal to k times the number whose digits are merely a cyclic permutation of its own leftmost digit to the right. %D A094676 H. Camous, Jouer Avec Les Maths, "Chassez le naturel", Section I, Problem 3 pp. 20;31-2, Les Editions D'Organisation, Paris 1984. %D A094676 L. A. Graham, Ingenious Mathematical Problems and Methods, "End At The Beginning", Problem 72 pp. 44;212-3 Dover NY 1959. %F A094676 n prepended to n*(10^m - n)/(10*n - 1), where m=A094224(n)-1. %e A094676 a(4)=410256=4*102564 %K A094676 fini,nonn %O A094676 2,1 %A A094676 Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 07 2004
- though the definition doesn't look right! I will edit it.
Anyway, could someone send me more terms of the sequence 210526315789473684, ... of numbers with the Freeman Dyson doubling property?
Neil
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (3)
-
Allan Wechsler -
Dave Blackston -
N. J. A. Sloane