A033023 -id:A033023 - cp's OEIS frontend

A374749 Number of zygodrome numbers in base-10 (A033023) with n or fewer digits.

0, 9, 18, 108, 279, 1260, 3780, 15129, 49158, 185328, 627759, 2295720, 7945560, 28607049, 100117098, 357580548, 1258634439, 4476859380, 15804569340, 56096303769, 198337427838, 703204161768, 2488241012319, 8817078468240, 31211247579120, 110564953793289

Offset: 1

Ruud H.G. van Tol, Sep 06 2024

Maarten Looijen, Over getallen gesproken - Talking about numbers, Van Haren Publishing, 2018, page 513.

Giovanni Resta, Zygodrome.
Index entries for linear recurrences with constant coefficients, signature (2,8,-9).

Cf. A015445, A033023.

LinearRecurrence[{1, 9}, {1, 10}, 26] - 1 (* Amiram Eldar, Sep 06 2024 *)

PARI
```
a(n) = ([0, 1; 9, 1]^n)[2, 2] - 1;
```

a(n) = A015445(n) - 1.
From Stefano Spezia, Sep 07 2024: (Start)
G.f.: 9*x^2/((1 - x)*(1 - x - 9*x^2)).
E.g.f.: exp(x/2)*(cosh(sqrt(37)*x/2) + sinh(sqrt(37)*x/2)/sqrt(37)) - exp(x). (End)

A014181 Numbers > 9 with all digits the same.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999, 11111, 22222, 33333, 44444, 55555, 66666, 77777, 88888, 99999, 111111, 222222, 333333, 444444, 555555, 666666, 777777, 888888

Offset: 1

N. J. A. Sloane

Original definition: Numbers in which all digits are repeated. (This would also include all terms of A033023 (e.g., 1100, 1122, ...) and maybe also 1010, 1212, etc.) - M. F. Hasler, Jun 24 2016

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, -10).

Same as A010785 except for one-digit numbers.

Table[Map[FromDigits@ Table[#, k] &, Range@ 9], {k, 2, 6}] // Flatten (* Michael De Vlieger, Jun 24 2016 *)

A014181(n)=10^((n+17)\9)\9*((n-1)%9+1) \\ See A010785 for a nxt() function. - M. F. Hasler, Jun 24 2016

Better definition and offset changed to 1 by M. F. Hasler, Jun 24 2016

A115853 Numbers where every digit that is present occurs more than once (not necessarily consecutively).

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1001, 1010, 1100, 1111, 1122, 1133, 1144, 1155, 1166, 1177, 1188, 1199, 1212, 1221, 1313, 1331, 1414, 1441, 1515, 1551, 1616, 1661, 1717, 1771, 1818, 1881, 1919, 1991, 2002

Offset: 1

Franklin T. Adams-Watters, Mar 13 2006

Through terms shown, this is also numbers where every digit present occurs the same number of times (not necessarily consecutively). The first number in this sequence not in that one is 10001.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A033023, A014181.

filter:= proc(n)
local L,S;
L:= convert(n,base,10);
S:= convert(L,set);
min(seq(numboccur(i,L),i=S)) > 1
end proc:
select(filter, [$1..10000]); # Robert Israel, May 28 2014

edQ[n_]:=Min[Select[DigitCount[n],#!=0&]]>1; Select[Range[2100],edQ] (* Harvey P. Dale, Dec 13 2018 *)

A079035 Squares, not ending in zero, in which all the digits occur in groups of 2 or more.

Original entry on oeis.org

7744, 5500002244, 11122233444, 8844449977444, 992255544007744, 5599881199880003344, 22233777661111993344, 448855114422552277444, 6633366633992211177744, 8882255222228844776644, 8888224466776699333444, 663336644555994400033344, 771133335566554442222244

Offset: 1

Sudipta Das (juitech(AT)vsnl.net), Feb 01 2003

			a(2) = 5500002244 because the square consists of a group of 2 5's, followed by a group of 4 0's, then 2 2's and finally 2 4's.

Giovanni Resta, Table of n, a(n) for n = 1..76 (terms < 10^38)
Frank Rubin, Repeated Digits

Cf. A079036, A033023.

More terms from Giovanni Resta, Sep 06 2018

A079036 Squares of these numbers have all their digits occurring in groups of 2 or more, without ending in zero.

Original entry on oeis.org

88, 74162, 105462, 2973962, 31500088, 2366406812, 4715270688, 21186201038, 81445482588, 94245717262, 94277380462, 814454814312, 878141979162, 2579792842738, 3339943529162, 5744665119912, 10490710229162, 10546200105462, 10950910657262, 23577201339312

Offset: 1

Sudipta Das (juitech(AT)vsnl.net), Feb 01 2003

A larger example is given by 939269775530142412^2 = 882227711224444113355445577001177744. The sequence is infinite, since it contains the numbers (10^k+1)*74162 and (10^k+1)*105462 for any k>=13. - Giovanni Resta, Sep 20 2006

			a(2) = 74162 because 74162^2 = 5500002244 consists of a group of 2 5's, followed by a group of 4 0's, then 2 2's and finally 2 4's.

Giovanni Resta, Table of n, a(n) for n = 1..76 (terms < 10^19)
Frank Rubin, Repeated Digits.

Cf. A079035, A033023.

More terms from Giovanni Resta, Sep 20 2006

More terms from Giovanni Resta, Sep 06 2018

A321538 a(n) = A321536(A321537(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 33, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 44, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 66, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 77, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 88, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 99, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 111, 11, 11, 11, 11, 11, 11, 11, 11, 0, 0, 22, 0

Offset: 0

N. J. A. Sloane, Nov 13 2018

More than the usual number of terms are shown in order to reach some interesting terms.

Fixed points correspond to 0 alongside A033023. - Rémy Sigrist, Nov 16 2018

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A033023, A321536, A321537.

A base-10 analog of A318925.

Delete a digit A in n if it either forms a run of length 1 or begins a substring AwAA, where w consists of runs of length 1. - Charlie Neder, Dec 16 2018

A116058 n times the n-th prime is made of nontrivial runs of identical digits.

Original entry on oeis.org

5, 12, 19822, 281984, 548600, 1665863, 6154167, 668316881, 705145645, 4259192969, 6791367165, 13266297467

Offset: 1

Giovanni Resta, Feb 13 2006

A run of length 1 is trivial.

			p(4259192969)=103577289271 and
4259192969*103577289271=441155662211122335599.

Cf. A033023, A116059, A116060.

A116059 The prime p times its index gives a number that is made of nontrivial runs of identical digits.

Original entry on oeis.org

11, 37, 222553, 3981707, 8141131, 26706079, 107244499, 14956328279, 15820183739, 103577289271, 168466166087, 338351202499

Offset: 1

Giovanni Resta, Feb 13 2006

A run of length 1 is trivial.

			p(4259192969)=103577289271 and
4259192969*103577289271=441155662211122335599.

Cf. A033023, A116058, A116060.

A116060 Numbers made of nontrivial runs of identical digits obtained multiplying a number m by the m-th prime.

Original entry on oeis.org

55, 444, 4411445566, 1122777666688, 4466224466600, 44488668881177, 660000556677333, 9995566666633377799, 11155533666655666655, 441155662211122335599, 1144115588776688333355, 4488667700668887770033

Offset: 1

Giovanni Resta, Feb 13 2006

A run of length 1 is trivial.

			p(4259192969)=103577289271 and
4259192969*103577289271=441155662211122335599.

Cf. A033023, A116058, A116059.

Select[Table[m*Prime[m],{m,10^6}],Min[Length/@Split[IntegerDigits[#]]]>1&] (* The program generates the first 5 terms of the sequence. *) (* Harvey P. Dale, Jun 05 2024 *)

A116064 Powerful numbers (A001694) made of nontrivial runs of identical digits.

Original entry on oeis.org

7744, 665500, 774400, 5556600, 6655000, 7744000, 55566000, 66550000, 77440000, 447722888, 448855888, 555660000, 665500000, 774400000, 1155775544, 5500002244, 5511005500, 5556600000, 6655000000, 7744000000, 11000004488, 11122233444, 22244466888

Offset: 1

Giovanni Resta, Feb 13 2006

A run of length 1 is trivial.

			448855888 = 2^4*13^3*113^2.

Amiram Eldar, Table of n, a(n) for n = 1..201 (terms below 10^18)
Index entries for sequences related to powerful numbers.

Intersection of A001694 and A033023.

a(16)-a(23) from Donovan Johnson, Jun 29 2011

cp's OEIS Frontend

A374749 Number of zygodrome numbers in base-10 (A033023) with n or fewer digits.

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A014181 Numbers > 9 with all digits the same.

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A115853 Numbers where every digit that is present occurs more than once (not necessarily consecutively).

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A079035 Squares, not ending in zero, in which all the digits occur in groups of 2 or more.

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A079036 Squares of these numbers have all their digits occurring in groups of 2 or more, without ending in zero.

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A321538 a(n) = A321536(A321537(n)).

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A116058 n times the n-th prime is made of nontrivial runs of identical digits.

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A116059 The prime p times its index gives a number that is made of nontrivial runs of identical digits.

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A116060 Numbers made of nontrivial runs of identical digits obtained multiplying a number m by the m-th prime.

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