A069871 -id:A069871 - cp's OEIS frontend

A077192 {Concatenation of n-1 and n+1}/n where n is a member of A069871.

8, 90, 92, 98, 992, 998, 9992, 9998, 99992, 99998, 999992, 999994, 999998, 9999992, 9999998, 99999992, 99999998, 999999992, 999999998, 9999999992, 9999999998, 99999999992, 99999999998, 999999999992, 999999999994, 999999999998, 9999999999992

Offset: 1

Amarnath Murthy, Nov 01 2002

			8 and 90 are members since 24/3 = 8, 810/9 = 90.
999994 is in sequence because 999994 = 142856142858/142857 and 142857 is a member of A069871.

Cf. A069871.

More terms from Sascha Kurz, Feb 10 2003
Inserted missing terms a(12) and a(25) by Paolo P. Lava, Nov 05 2014
Offset changed by Sean A. Irvine, May 09 2025

A069860 Numbers n that divide the concatenation of n+1 and n+2.

Original entry on oeis.org

1, 2, 3, 4, 6, 17, 34, 51, 167, 334, 501, 1667, 3334, 5001, 14286, 16667, 33334, 50001, 166667, 333334, 500001, 1666667, 3333334, 5000001, 16666667, 33333334, 50000001, 166666667, 333333334, 500000001, 1666666667, 3333333334

Offset: 1

Amarnath Murthy, Apr 18 2002

			17 divides 1819 hence is a member.

Cf. A069860, A069861, A069862, A069871, A088797, A088798.

Select[ Range[10^7], Mod[ FromDigits[ Join[ IntegerDigits[ # + 1], IntegerDigits[ # + 2]]], # ] == 0 & ]

For k > 0, sequence contains (10^k+2)/2, (10^k+2)/3, (10^k+2)/6 and (10^(6k-1)+2)/7. The only other terms are 1 and 3. - David Wasserman, Aug 25 2005

Edited and extended by Robert G. Wilson v, Apr 22 2002

More terms from David Wasserman, Aug 25 2005

A088797 Numbers n > 2 such that n divides the concatenation of n-2 and n-1.

Original entry on oeis.org

3, 7, 67, 667, 6667, 66667, 666667, 2857143, 6666667, 66666667, 666666667, 1052631579, 6666666667, 66666666667, 666666666667, 2857142857143, 6666666666667, 11764705882353, 66666666666667, 666666666666667

Offset: 1

Chuck Seggelin, Oct 18 2003

For a 10-digit number, the difference between cat((n+2),(n+1)) and cat((n-2),(n-1)) is 40000000002 (as long as n-2 to n+2 are all numbers with the same number of digits). This difference has only 3 divisors which are ten digits long (1052631579, 2105263158 and 6666666667) of which two belong to the sequence. As 40000000002 has no other 10-digit factors, it is necessary to consider 11-digit numbers to obtain more terms.
From Robert G. Wilson v, Oct 21 2003, Oct 28 2003, Sep 23 2015 & Oct 24 2015: (Start)
All numbers of the forms
2(10^n-1)/3 + 1,
floor(2(10^(6n + 1) - 1)/7 + 1),
floor(2(10^(16n - 2) - 1)/17 + 1), and
floor(2(10^(18n - 8) - 1)/19 + 1), for n > 0 are members.
The only term not one of the above forms so far is 3. But it is included when n=0 for the second form.
(End)
If numbers less than 3 are acceptable, then an argument could be made that 1 is a terms since cat(n-2,n-1) is -10 which is == 0 (mod 1). - Robert G. Wilson v, Sep 29 2015
From Robert Israel, Oct 18 2015: (Start)
Numbers n of the form (2*10^m + 1)/k where k = 3, or k = 7 and m == 1 mod 6, or k = 17 and m == 14 mod 16, or k = 19 and m == 10 mod 18.
This is because n | (n-2)*10^m + (n-1) iff n | 2*10^m + 1.
But since we need 10^m >= n > 10^(m-1), 2*10^m+1 = k*n where 3 <= k <= 20.
The only numbers in that range that ever divide 2*10^m+1 are 3,7,17,19. (End)

			a(2)=7 because (7-2) concatenated with (7-1) yields 56 and 7 is a divisor of 56.
a(4)=667 because 667 is a divisor of 665666.
.

Robert Israel, Table of n, a(n) for n = 1..1285 (n = 1..104 from Robert G. Wilson v)

Cf. A069860, A069862, A069871, A088798.

M:= 20: # to get all terms with <= M digits
select(type, [seq(seq((2*10^m+1)/k, k=[19,17,7,3]),m=1..M)],integer); # Robert Israel, Oct 18 2015

Select[ Range[8250000000], Mod[ FromDigits[ Join[ IntegerDigits[ # - 2], IntegerDigits[ # - 1]]], # ] == 0 &]
fQ[n_] := Mod[ FromDigits[ Join[ IntegerDigits[n - 2], IntegerDigits[n - 1]]], n] == 0; k = 1; lst = {}; Select[ Flatten@ Table[ Select[ Divisors[4*10^n + 2], 10^(n - 1) < # < 10^n &], {n, 15}], fQ] (* Robert G. Wilson v, Sep 05 2015 *)

for(n=3, 1e6, if((k=eval(Str(n-2,n-1))) && k % n == 0, print1(n", "))) \\ Altug Alkan, Sep 25 2015

Extended by Robert G. Wilson v, Oct 21 2003

Further terms from Chuck Seggelin, Oct 29 2003

A088798 Numbers n that divide the concatenation of n-1, n-2 and n-3.

Original entry on oeis.org

3, 19, 57, 3276457, 9829371, 22997937, 24687460011, 24504559526049, 1152870338086169, 3458611014258507, 19831522709797616449, 54128285729329681609, 59494568129392849347, 61582096835687335289

Offset: 1

Chuck Seggelin, Oct 19 2003

Each member of this sequence also appears to be a divisor of the number formed when concatenating (n+1), (n+2) and (n+3) in that order. Each nonprime member of the terms above appears to be divisible by 3. Further note that apart from 3 itself, if a(n) is a prime, then 3 * a(n) also appears to be a member. 19*3=57, 3276457*3=9829371. More prime members would need to be found to test this.

			a(2)=19 because 19 is a divisor of 181716. a(4)=3276457 because 3276457 is a divisor of 327645632764553276454.

Cf. A069860, A069862, A069871, A088797.

prevcatOld := proc(n,t,o) local i,s; s := ""; for i from 1 to t do if o="a" then s := cat(convert(n-i,string),s) else s := cat(s,convert(n-i,string)) fi; od; parse(s) end; nextdivcat := proc(startAt,endAt,numTerms,catOrder) local i; for i from startAt to endAt while (prevcatOld(i,numTerms,catOrder) mod i > 0) do od; if i<=endAt then i else -1 fi; end; s := NULL; t := 2; for j from 1 to 10 do t := nextdivcat(t+1,23000000,3,"d"); s := s,t od; print(s);

Do[ If[ Mod[ FromDigits[ Join[ IntegerDigits[2n], IntegerDigits[2n - 1], IntegerDigits[2n - 2]]], (2n + 1)] == 0, Print[2n + 1]], {n, 1, 700000000}]

Edited by Robert G. Wilson v, Oct 20 2003

More terms from David Wasserman, Aug 25 2005

A088877 Numbers k that are divisors of the number formed by concatenating (k-1), k and (k+1), in that order.

Original entry on oeis.org

1, 3, 9, 11, 33, 111, 117, 143, 189, 231, 259, 273, 297, 333, 351, 407, 429, 481, 693, 777, 819, 1111, 1233, 1507, 2409, 3333, 4521, 7227, 7373, 11111, 26829, 27273, 33333, 81819, 100899, 101101, 108911, 111111, 123321, 128713, 129987, 142857

Offset: 1

Chuck Seggelin, Oct 20 2003

The sequence of numbers that divide the concatenation of k-1 and k+1 (A069871) is a subsequence of this sequence. - David Wasserman, Mar 26 2004

			a(7)=117 because 117 is a divisor of 116117118.

Paolo P. Lava, Table of n, a(n) for n = 1..100

Cf. A069871.

Select[Range[143000],MemberQ[Divisors[FromDigits[Flatten[ IntegerDigits/@ {#-1,#,#+1}]]],#]&] (* Harvey P. Dale, Aug 27 2020 *)

A249647 Numbers n that divide the concatenation of n+1 and n-1.

Original entry on oeis.org

1, 3, 9, 11, 33, 99, 111, 333, 999, 1111, 3333, 9999, 11111, 33333, 99999, 111111, 142857, 333333, 999999, 1111111, 3333333, 9999999, 11111111, 33333333, 99999999, 111111111

Offset: 1

Paolo P. Lava, Nov 03 2014

A069871 is a subset of this sequence.
All the numbers of the form (10^k - 1)/9, (10^k - 1)/3 and (10^k-1) are members.
Any concatenation of 142857 with itself is part of the sequence, too.
All integers of the form (10^d-1)/k where 1 <= k <= 9. - Robert Israel, Jan 06 2016
It appears that equivalent definitions are: (A) Numbers which divide the repunit of the same length, n | R(length(n)). (B) Numbers equal to one or more concatenations of one among {1, 3, 9, 142857}. Is there a proof for this? - M. F. Hasler, Jun 10 2016

			3 belongs to this sequence as 3 divides 42, 11 belongs to this sequence as 11 divides 1210.
9 belongs to this sequence as 9 divides the concatenation of 10 and 8, i.e., 108.

Robert Israel, Table of n, a(n) for n = 1..3163

Cf. A069871.

with(numtheory): P:=proc(q) local n;
for n from 1 to q do if type(((n+1)*10^(ilog10(n-1)+1)+n-1)/n,integer) then print(n);
fi; od; end: P(10^15);
# alternative:
seq(op(select(type, [seq((10^d-1)/k, k=[9,7,3,1])],integer)),d=1..20); # Robert Israel, Jan 06 2016

Select[Range[2, 10^7], Divisible[FromDigits[IntegerDigits[# + 1]~Join~IntegerDigits[# - 1]], #] &] (* Michael De Vlieger, Jan 06 2016 *)

for(n=1,10^8, s=concat(Str(n+1),Str(n-1));if(!(eval(s)%n),print1(n,", "))) \\ Derek Orr, Nov 03 2014

a(n) = 1000001*a(n-19) + 1000000*a(n-38). - Robert Israel, Jan 06 2016

a(23)-a(25) from Derek Orr, Nov 03 2014

A281232 Numbers k such that k+2 divides concat(k, k+1).

Original entry on oeis.org

1, 5, 65, 665, 6665, 66665, 666665, 2857141, 6666665, 66666665, 666666665, 1052631577, 6666666665, 66666666665, 666666666665, 2857142857141, 6666666666665, 11764705882351, 66666666666665, 666666666666665, 6666666666666665, 66666666666666665, 666666666666666665

Offset: 1

Paolo P. Lava, Jan 18 2017

Numbers of the form 60*(10^j - 1)/9 + 5, for j >= 0, belong to the sequence.
The ratios are: 4, 8, 98, 998, 9998, 99998, 999998, 9999994, 9999998, 99999998, 999999998, 9999999982, 9999999998, ...
Numbers of the form t(j) = 20*(10^(6*j) - 1)/7 + 1, for j >= 0, belong to the sequence, because (10^(6*j+1)*t(j) + t(j) + 1)/(t(j) + 2) = 10^(6*j+1) - 6. - Bruno Berselli, Oct 09 2018

			concat(2857141, 2857142) / 2857143 = 28571412857142 / 2857143 = 9999994.

Cf. A088797, A281233, A069871.

with(numtheory): P:=proc(q) local c,n;
for n from 1 to q do c:=n*10^(ilog10(n+1)+1)+n+1;
if type(c/(n+2),integer) then print(n); fi; od; end: P(10^9);

Select[Range[10^7], Divisible[FromDigits@ Flatten@ Map[IntegerDigits, {#, # + 1}], # + 2] &] (* Michael De Vlieger, Jan 19 2017 *)

isok(n) = !(eval(Str(n, n+1)) % (n+2)); \\ Michel Marcus, Oct 09 2018

a(n) = A088797(n) - 2. - Alois P. Heinz, Jan 19 2017

More terms from Alois P. Heinz, Jan 19 2017

A281233 Numbers k such that k-1 | concat(k, k+1).

Original entry on oeis.org

2, 3, 4, 5, 7, 18, 35, 52, 168, 335, 502, 1668, 3335, 5002, 14287, 16668, 33335, 50002, 166668, 333335, 500002, 1666668, 3333335, 5000002, 16666668, 33333335, 50000002, 166666668, 333333335, 500000002, 1666666668, 3333333335

Offset: 1

Paolo P. Lava, Jan 18 2017

Numbers of the form 10^(j+1) + 60*(10^j-1)/9 + 8, 30*(10^j-1)/9 + 5 and 5*10^j + 2, for j>=0, belong to the sequence.

The ratios are: 23, 17, 15, 14, 13, 107, 104, 103, 1007, 1004, 1003, 10007, 10004, 10003, 100008, 100007, 100004, 100003, 1000007, 1000004, 1000003, 10000007, 10000004, 10000003, 100000007, 100000004, 100000003, 1000000007, 1000000004, 1000000003, ...

			concat(2,3) / 1 = 23 / 1 = 23; concat(3,4) / 2 = 34 / 2 = 17; etc.

Cf. A069860, A281232, A069871.

with(numtheory): P:=proc(q) local c,n;
for n from 2 to q do c:=n*10^(ilog10(n+1)+1)+n+1;
if type(c/(n-1),integer) then print(n); fi; od; end: P(10^9)

Select[Range[2, 10^7], Divisible[FromDigits@ Flatten@ Map[IntegerDigits, {#, # + 1}], # - 1] &] (* Michael De Vlieger, Jan 19 2017 *)

a(n) = A069860(n) + 1. - Alois P. Heinz, Jan 19 2017

Two more terms (using A069860) from Alois P. Heinz, Jan 19 2017

A292885 a(n) is the least number k such that k | concat(k-n,k-n+1,…,k,…,k+n-1,k+n).

Original entry on oeis.org

1, 3, 3, 9, 6, 9, 9, 9, 9, 27, 15, 33, 18, 27, 18, 21, 18, 27, 27, 27, 24, 27, 27, 27, 27, 27, 27, 143, 34, 143, 45, 63, 36, 39, 39, 45, 42, 143, 89, 57, 45, 43, 143, 99, 54, 135, 154, 63, 63, 63, 75, 63, 154, 189, 66, 165, 72, 171, 153, 189, 90, 63, 81, 69, 69

Offset: 0

Paolo P. Lava, Sep 26 2017

			a(4) = 6 because concat(2, 3, 4, 5, 6, 7, 8, 9, 10) = 2345678910 is a multiple of 6 and 6 is the least number to have this property.

Paolo P. Lava, Table of n, a(n) for n = 0..500

Cf. A069860, A069862, A069871, A088797, A088798, A088877.

P:=proc(q) local a,j,k,n; for k from 1 by 2 to q do for n from 1 to q do a:=n;
for j from 1 to k-1 do a:=a*10^(ilog10(n+j)+1)+n+j; od;
if type(a/(n+(k-1)/2),integer) then print(n+(k-1)/2); break; fi; od; od; end: P(10^9);

Table[SelectFirst[Range[10^3], Function[k, If[And[FreeQ[#, ?(# < 0 &)], First@ # != 0], Divisible[FromDigits@ Flatten@ Map[IntegerDigits, #], k]] &[k + Range[-n, n]] ] ], {n, 0, 64}] (* _Michael De Vlieger, Sep 27 2017 *)

isok(k, n) = {my(s = ""); for (i=0, 2*n, s = concat(s, k-n+i);); (eval(s) % k) == 0;}
a(n) = {my(k = n+1); while(!isok(k,n), k++); k;} \\ Michel Marcus, Oct 06 2017

cp's OEIS Frontend

A077192 {Concatenation of n-1 and n+1}/n where n is a member of A069871.

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A069860 Numbers n that divide the concatenation of n+1 and n+2.

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A088797 Numbers n > 2 such that n divides the concatenation of n-2 and n-1.

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A088798 Numbers n that divide the concatenation of n-1, n-2 and n-3.

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A088877 Numbers k that are divisors of the number formed by concatenating (k-1), k and (k+1), in that order.

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A249647 Numbers n that divide the concatenation of n+1 and n-1.

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A281232 Numbers k such that k+2 divides concat(k, k+1).

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A281233 Numbers k such that k-1 | concat(k, k+1).