A108551 -id:A108551 - cp's OEIS frontend

A234512 Numbers n = d(0)d(1)d(2)...d(r) such that d(i) is the number of differences |d(i)-d(i-1)| equal to i in n, i = 1,2,...,r.

110, 311000, 2301000, 3003000, 3120000, 42100000, 410300000, 430100000

Offset: 1

Michel Lagneau, Dec 27 2013

In the decimal system a differential autobiographical number is a natural number such that d(0) is the number of differences |d(i)-d(i-1)| = 0, d(1) is the number of differences |d(i)-d(i-1)| = 1, and so on.
Property of this sequence: the sum of the decimal digits of a(n) equals length(a(n))-1.
It is possible to extend this problem by counting the differences |d(i)-d(i-1)| with the additional difference |d(r)-d(1)|. So we find a new sequence b(n) = 22100, 311100, 3022000, 20402000, 31310000, 40004000, 422010000, 430110000 with the property that the sum of the decimal digits of b(n) equals length(b(n)).

			311000 is in the sequence because the differential digits are:
|1-3| = 2;
|1-1| = 0;
|0-1| = 1;
|0-0| = 0;
|0-0| = 0, and
0 appears three times => 3;
1 appears one time => 1;
2 appears one time  => 1;
3 appears zero time => 0;
4 appears zero time => 0;
5 appears zero time => 0, hence a(2) = 311000.

Tanya Khovanova, Autobiographical Numbers

Cf. A037904, A046043, A108551, A138480.

with(numtheory):for n from 10 to 10^10 do:T:=array(0..9):for k from 0 to 9 do:T[k]:=0:od:x:=convert(n,base,10):n1:=nops(x):for i from 1 to n1-1 do:a:=abs(x[i]-x[i+1]):T[a]:=T[a]+1:od:s:=sum('T[i]*10^(10-i-1)','i'=0..9): for u from 9 by -1 to 1 do:if T[0]<>0 and irem(s,10^u)=0 and s/10^u = n then print(n):else fi:od:od:

A274943 Smallest self-descriptive number in base b, or -1 if no such number exists.

Original entry on oeis.org

-1, -1, 100, 1425, -1, 389305, 8946176, 225331713, 6210001000, 186492227801, 6073061476032, 213404945384449, 8054585122464440, 325144322753909625, 13983676842985394176

Offset: 2

N. J. A. Sloane, Jul 23 2016, following an email from Amarnath Krishnamurthy [Amarnath Murthy] and in addition using data from A108551

A self-descriptive number in base b has b digits, indexed by 0 ... b-1 and for all n, the n-th digit equals the number of n's in the number. In base 10 there is exactly one such number, 6210001000.

			1210_4 = 100, 21200_5 = 1425, 3211000_7 = 389305,
42101000_8 = 8946176, 521001000_9 = 225331713, 6210001000_10,
72100001000_11 = 186492227801, 821000001000_12 = 6073061476032,
9210000001000_13 = 213404945384449, (10)2100000001000_14 =
8054585122464440, (11)21000000001000_15 = 325144322753909625,
(12)21000000001000_16 = 13983676842985394176, etc.

Very similar to A108551.

A282535 a(n) is the maximum number of "describing"-steps for an n-chain before entering a loop.

Original entry on oeis.org

3, 4, 7, 4, 7, 7, 7, 6, 7, 6, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 4

Felix Fröhlich, Feb 17 2017

Conjecture: The size of terms of this sequence is unbounded (cf. Marichal, 2007, Corollary 5).

Jean-Luc Marichal, On perfect, amicable, and sociable chains, arXiv:0708.1491 [math.CO], 2007.

Cf. A046043, A108551.

pad(d, n) = while(#d != n, d = concat([0], d)); d;
say(d, n) = vector(n, k, sum(j=1, #d, d[j] == (k-1)));
isok(v, n) = my(vs = vecsort(v,,8)); (#vs > 1) && (#vs Michel Marcus, Feb 25 2017

Name edited by Michel Marcus, Feb 26 2017

cp's OEIS Frontend

A234512 Numbers n = d(0)d(1)d(2)...d(r) such that d(i) is the number of differences |d(i)-d(i-1)| equal to i in n, i = 1,2,...,r.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A274943 Smallest self-descriptive number in base b, or -1 if no such number exists.

Views

Author

Keywords

Comments

Examples

Crossrefs

A282535 a(n) is the maximum number of "describing"-steps for an n-chain before entering a loop.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions