A069862 -id:A069862 - cp's OEIS frontend

A088343 Records in A069862.

1, 2, 5, 9, 10, 22, 25, 26, 110, 998, 1158, 1410, 9860, 100270, 999100, 1005274, 1007044, 1055274, 1059163, 1063242, 1065027, 1083148, 1099446, 1103722, 1144506, 10146315, 99992456, 99997120, 101009312, 101924112, 102163463, 102706016

Offset: 1

N. J. A. Sloane, Nov 14 2003

A069862 is the smallest k such that n divides the concatenation of numbers from (n+1) to (n+k), where (n+1) is on the most significant side. - Robert G. Wilson v, Nov 14 2003

Chuck Seggelin, Concatenation of Consecutive Integers.

Cf. A069862, A088947.

f[n_] := Block[{k = n + 1}, d = k; While[ d != 0, k++; d = Mod[d*10^Floor[ Log[10, k] + 1] + k, n]]; k - n]; a = 0; Do[b = f[n]; If[b > a, a = b; Print[n, " = ", b]], {n, 1, 34500}]

More terms from Robert G. Wilson v and Chuck Seggelin (chuck(AT)plastereddragon.com), Nov 14 2003
Edited by Charles R Greathouse IV, Apr 30 2010
a(29)-a(32) from Michael S. Branicky, Jul 29 2024

A088947 Positions of records in A069862.

Original entry on oeis.org

1, 2, 5, 7, 10, 13, 25, 27, 31, 37, 151, 277, 385, 451, 902, 6314, 6601, 16627, 24673, 30371, 31451, 32338, 36058, 41173, 44083, 44215, 47322, 52580, 558304, 659879, 744007, 909634

Offset: 1

Robert G. Wilson v, Oct 29 2003

A069862 is the smallest k such that n divides the concatenation of numbers from (n+1) to (n+k), where (n+1) is on the most significant side.

			f(31) = 110 which exceeds f(27) = 26; f(451) = 100270; f(902) = 999100.

Chuck Seggelin, Concatenation of Consecutive Integers.

Cf. A069862, A088343 (for records themselves).

f[n_] := Block[{k = n + 1}, d = k; While[ d != 0, k++; d = Mod[d*10^Floor[ Log[10, k] + 1] + k, n]]; k - n]; a = 0; Do[b = f[n]; If[b > a, a = b; Print[n, " = ", b]], {n, 1, 34500}]

More terms from Robert G. Wilson v and Chuck Seggelin (chuck(AT)plastereddragon.com), Nov 14 2003
Edited by Charles R Greathouse IV, Apr 24 2010
a(29)-a(32) from Michael S. Branicky, Jul 29 2024

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

A069871 Numbers m that divide the concatenation of m-1 and m+1.

Original entry on oeis.org

3, 9, 11, 33, 111, 333, 1111, 3333, 11111, 33333, 111111, 142857, 333333, 1111111, 3333333, 11111111, 33333333, 111111111, 333333333, 1111111111, 3333333333, 11111111111, 33333333333, 111111111111, 142857142857, 333333333333, 1111111111111

Offset: 1

Amarnath Murthy, Apr 24 2002

All the numbers of the form (10^k - 1)/3 and (10^k - 1)/9 are terms.
These (i.e., (10^k - 1)/3 for k >= 1, (10^k - 1)/9 for k >= 2, and (10^(6*k) - 1)/7 for k >= 1) are all the terms of the sequence, apart from 9.  This is because if 10^(i-1) <= x+1 < 10^i, x | 10^i*(x-1) + x + 1 iff x | 10^i - 1, and then 1 < d = (10^i - 1)/x <= (10^i - 1)/(10^(i-1)-1) < 10.  Since 2,4,5,6,8 can't divide 10^i-1, d must be 3, 7 or 9. - Robert Israel, Nov 04 2014
The terms of the sequence satisfy the condition that both m-1 and m+1 must be greater than 0. If m-1=0 were admitted then 1 would also be part of the sequence. - Michel Marcus, Nov 05 2014

			3 belongs to this sequence since 3 divides 24; 11 belongs to this sequence since 11 divides 1012.
9 belongs to this sequence since 9 divides the concatenation of 8 and 10, i.e., 810.
142857 belongs to this sequence since 142857 divides the concatenation of 142856 and 142858, i.e., 142856142858/142857 = 999994.

Cf. A069860, A069862, A088797, A088798.

Cf. A077192.

N:= 10: # to get all terms with at most N digits
3,9, seq(seq((10^k-1)/d, d = `if`(k mod 6 = 0, [9,7,3],[9,3])), k = 2 .. N); # Robert Israel, Nov 04 2014

Select[ Range[10^8], Mod[ FromDigits[ Join[ IntegerDigits[ # - 1], IntegerDigits[ # + 1]]], # ] == 0 & ]

isok(n) = eval(concat(Str(n-1), Str(n+1))) % n == 0; \\ Michel Marcus, Nov 04 2014

From Robert Israel, Nov 04 2014: (Start)
a(1+2*j + 13*k) = (10^(1+j+6*k)-1)/9, j=0..5, k >= 0 (except for j=k=0).
a(2*j + 13*k) = (10^(j+6*k)-1)/3, j=0..5, k >= 0 (except for j=k=0 and j=1,k=0).
a(13*k - 1) = (10^(6*k)-1)/7, k >= 1.
(End)

More terms from Sascha Kurz, Feb 10 2003

Missing a(12) added by Paolo P. Lava and missing a(25) added by Alois P. Heinz, Nov 03 2014

A069861 Numbers n that divide the concatenation of next n numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 10, 11, 12, 15, 18, 20, 21, 22, 25, 27, 30, 33, 36, 39, 42, 44, 45, 50, 54, 56, 60, 68, 72, 75, 90, 98, 100, 104, 108, 110, 111, 114, 120, 125, 126, 130, 132, 135, 140, 148, 150, 154, 155, 156, 162, 168, 180, 182, 185, 196, 198, 200, 205, 210, 216

Offset: 1

Amarnath Murthy, Apr 18 2002

			9 divides 101112131415161718 hence is a member.

Cf. A069860, A069862.

f[n_] := Block[{j = 0, k = n + 1}, While[k < 2n + 1, j = ToExpression[ StringJoin[ ToString[j], ToString[k]]]; k++ ]; j]; Select[ Range[300], IntegerQ[ f[ # ] / # ] & ]

Edited, corrected and extended by Robert G. Wilson v

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

A069863 a(n) is the shortest concatenation of two or more decreasing consecutive numbers (n)(n-1)...(0) such that n divides a(n), or n if there is no such concatenation.

Original entry on oeis.org

10, 210, 321, 432, 543210, 654, 7, 8765432, 987654321, 109876543210, 11, 1211109876, 13, 14131211109876, 151413121110, 1615141312, 171615, 181716151413121110, 19181716, 20, 21, 2221201918171615141312, 23

Offset: 1

Amarnath Murthy, Apr 18 2002

a(n) = n for the following values of n: 7, 11, 13, 20, 21, 23, 25, 26, 29, 31, 37, 38, 39, 50, 52, 53, 58, 59, 62, 63, 65, 67, 71, 74, 81, 82, 85, 89, 91, 92, 95, 97, 100, 101, 106, 115, 116, 118, 130, 133, 145, 146, 152, 159, 169, 173, 175, 178, 182, 184, 185, 195, 196, 200, 212, 213, 214, 215, 217, 220, 221, 224, 227, 229, 230, 232, 235, 236, 239, 243, 250, ...

			a(7) = 7 the corresponding k = 0, a(2) =210, k=2. a(4) = 432, k = 2.

Cf. A069860, A069861, A069862.

f[n_] := Block[{j = n, k = n - 1}, While[j = ToExpression[ StringJoin[ ToString[j], ToString[k]]]; !IntegerQ[j/n] && k > -1, k-- ]; If[k == -1, Return[n], Return[j]]]; Table[f[n], {n, 1, 30}]

Edited by Robert G. Wilson v, Apr 22 2002

Edited by T. D. Noe, Apr 09 2009

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A088343 Records in A069862.

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A088947 Positions of records in A069862.

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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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A069871 Numbers m that divide the concatenation of m-1 and m+1.

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A069861 Numbers n that divide the concatenation of next n numbers.

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

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