A069860 -id:A069860 - cp's OEIS frontend

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

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

A069862 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.

Original entry on oeis.org

1, 2, 2, 2, 5, 2, 9, 4, 8, 10, 10, 8, 22, 16, 5, 4, 2, 8, 3, 20, 20, 10, 17, 12, 25, 22, 26, 16, 25, 20, 110, 20, 11, 2, 20, 8, 998, 52, 38, 20, 60, 20, 4, 32, 35, 42, 50, 20, 96, 50, 2, 96, 93, 26, 10, 20, 3, 50, 44, 20, 46, 40, 45, 40, 50, 32, 86, 32, 17, 20, 75, 72, 26, 926, 50

Offset: 1

Amarnath Murthy, Apr 18 2002

Minimum number of consecutive subsequent integers after n that must be concatenated together in ascending order such that n divides the concatenated term.

Concatenation always begins at n+1. Note that multiples of 11 seems to require more terms than any other number. 385 requires 9860. 451 requires 100270 terms be concatenated together into a 495,000 digit number. - Chuck Seggelin, Oct 29 2003; corrected by Chai Wah Wu, Oct 19 2014

			a(7) = 9 as 7 divides 8910111213141516 the concatenation of numbers from 8(= 7+1) to 16 (= 7+9).
a(5) = 5 because 5 will divide the number formed by concatenating the 5 integers after 5 in ascending order (i.e. 678910). a(385) = 9860 because 385 will divide the concatenation of 386,387,388,...,10245.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
C. Seggelin, Concatenation of Consecutive Integers.

Cf. A069860, A069861, A088797, A088799, A088868, A088870, A088872, A088885.

Records are in A088947, A088343.

c[1] := 1:for n from 2 to 172 do k := 1:g := (n+k) mod n:while(true) do k := k+1:b := convert(n+k,base,10):g := (g*10^nops(b)+n+k) mod n: if((g mod n)=0) then c[n] := k:break:fi:od:od:seq(c[l],l=1..172);

f[n_] := Block[{k = n + 1}, d = k; While[ !IntegerQ[d/n], k++; d = d*10^Floor[Log[10, k] + 1] + k]; k - n]; Table[ f[n], {n, 1, 75}] (* Robert G. Wilson v, Nov 04 2003 *)

def A069862(n):
    nk, kr, r = n+1, 1, 1 if n > 1 else 0
    while r:
        nk += 1
        kr = (kr + 1) % n
        r = (r*(10**len(str(nk)) % n)+kr) % n
    return nk-n # Chai Wah Wu, Oct 20 2014

More terms from Sascha Kurz, Jan 28 2003

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

A088799 Numbers n which are divisors of the number formed by concatenating (n-3), (n-2) and (n-1) in that order.

Original entry on oeis.org

3, 11, 9491, 12258083, 36774249, 2159487563, 2561252691, 2723957777, 6478462689, 8171873331, 333351714587, 146217070005379, 438651210016137, 13919982618156833, 41759947854470499, 1278907806311980217974478364841

Offset: 1

Chuck Seggelin, Oct 19 2003

Apart from 11, each other term in this sequence appears to also be a factor of the number formed by concatenating (n+3), (n+2) and (n+1) in that order. All terms appear to be prime. When evaluating concat((n+3),(n+2),(n+1)) - concat((n-3),(n-2),(n-1)) for members larger than 11 the difference appears to always be a number of the form 6(0)...4(0)...2 with the same number of zeros on both sides of the 4. The member will be a prime factor of this number. By factoring numbers of the form 6(0)...4(0)...2 and testing the results, three further members of this sequence have been found: 2723957777, 1260049494294190236301929754269107568067 and 103945392111236434211250670719387720140245499. I have not included these in the list of members above as they were not arrived at through brute force as the first 4 terms were and there may be other intervening terms.

			a(3)=9491 because 9491 is a factor of 948894899490.

Cf. A069860, A088797, A088798.

More terms from David Wasserman, Aug 25 2005

A088800 Numbers n which are divisors of the number formed by concatenating (n-1), (n-2), (n-3) and (n-4) in that order.

Original entry on oeis.org

16, 43, 86, 3923, 7846, 1320532, 14869252, 37031789, 74063578, 1770748607, 3541497214, 7082994428, 50541770557, 1040474831558, 1076026363388, 2080949663116, 2746369613531, 3376243036861, 5492739227062, 6529386313577

Offset: 1

Chuck Seggelin, Oct 20 2003

Each member of this sequence appears to also be a factor of the number formed by concatenating (n+1), (n+2), (n+3) and (n+4) in that order. When evaluating concat((n+1),(n+2),(n+3),(n+4)) - concat((n-1),(n-2),(n-3),(n-4)) for members larger than 86 the difference appears to always be a number of the form 2(0)...4(0)...6(0)...8 with the same number of zeros following the 2, 4 and 6. The member will be a factor of this number. Further terms for the sequence can be produced by factoring numbers of this form. Let z=the number of zeros in one of the segments of a number d of the form 2(0)...4(0)...6(0)...8. Find the divisors of d. All divisors which are not of length z+1 are not members of this sequence and those that are of length z+1 are likely candidates and should be tested (note that apart from 16, candidates which are divisible by 8 appear to never be members). For example let d = 2000000000000000400000000000000060000000000000008. z=15. The divisors of d are numerous, but only one is z+1 (16) digits long: 7547657634163187. Testing this candidate confirms that it is also a member of this sequence.

			a(3)=86 because 86 is a factor of 85848382.

Cf. A069860, A088797, A088798, A088799.

A088868 Numbers n which are divisors of the number formed by concatenating (n-4), (n-3), (n-2) and (n-1) in that order.

Original entry on oeis.org

49, 109, 307121, 3837881, 415922011, 44886856951, 73071640562111, 11741452251865261, 138599925259848671

Offset: 1

Chuck Seggelin, Oct 20 2003

Each member of this sequence appears to also be a factor of the number formed by concatenating (n+4), (n+3), (n+2) and (n+1) in that order. When evaluating concat((n+4),(n+3),(n+2),(n+1)) - concat((n-4),(n-3),(n-2),(n-1)) for members of this sequence the difference appears to always be a number of the form 8(0)...6(0)...4(0)...2 with the same number of zeros following the 8, 6 and 4. The member will be a factor of this number. Terms for this sequence can be produced by factoring numbers of this form. Let z=the number of zeros in one of the segments of a number d of the form 8(0)...6(0)...4(0)...2. Find the divisors of d. All divisors which are not of length z+1 are not members of this sequence and those that are of length z+1 are possible candidates and should be tested. For example let d = 8000000000000000006000000000000000004000000000000000002. z=17. The divisors of d are numerous, but only two are z+1 (18) digits long: 138599925259848671 and 27719985051 9697342. Testing these candidates confirms that the first one is a member of this sequence.

No more terms < 10^29. - David Wasserman, Aug 26 2005

			a(2)=109 because 109 is a factor of 105106107108.

Cf. A069860, A088797, A088798, A088799, A088800.

A088870 Numbers n which are divisors of the number produced by concatenating (n-5), (n-4), ... (n-1) in that order.

Original entry on oeis.org

13677, 14647, 21291, 29567, 43941, 69031, 88701, 105991, 126507, 317973, 156304482823, 468913448469, 21729950852487, 2212933498428421, 6638800495285263, 12049739358792173, 36149218076376519, 11316117499289108644863

Offset: 1

Chuck Seggelin, Oct 20 2003

			a(1)=13677 because 13677 is a factor of 1367213673136741367513676.

Cf. A069860, A088797, A088798, A088799, A088800, A088868, A088869, A088871, A088872.

More terms from David Wasserman, Aug 26 2005

A088872 Numbers n which are divisors of the number produced by concatenating (n-10), (n-9), (n-8), ... (n-1) in that order.

Original entry on oeis.org

37, 1793, 12247, 13061, 12345679, 17675669, 59454523, 10035074789, 11207086103, 12174047803, 13606288721, 14036302277, 17989804447, 19541987747, 20070727459, 21841045129, 22431989513, 23140930781, 32217871691

Offset: 1

Chuck Seggelin, Oct 20 2003

			a(1)=37 because 37 is a factor of 27282930313233343536.

Cf. A069860, A088797, A088798, A088799, A088800, A088868, A088869, A088870, A088871.

More terms from David Wasserman, Aug 25 2005

A088869 Numbers n which are divisors of the number produced by concatenating (n-1), (n-2), ... (n-5) in that order.

Original entry on oeis.org

5, 15, 197, 591, 599, 985, 23363, 70089, 182415, 32533013, 97599039, 200814277564911, 334690462608185, 100208875079675883, 119690233947031245, 120541110603088381, 152213696825889183, 167014791799459805

Offset: 1

Chuck Seggelin, Oct 20 2003

			a(2)=15 because 15 is a factor of 1413121110.

Cf. A069860, A088797, A088798, A088799, A088800, A088868, A088870, A088871, A088872.

More terms from David Wasserman, Aug 26 2005

cp's OEIS Frontend

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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A069862 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.

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

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A088799 Numbers n which are divisors of the number formed by concatenating (n-3), (n-2) and (n-1) in that order.

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A088800 Numbers n which are divisors of the number formed by concatenating (n-1), (n-2), (n-3) and (n-4) in that order.

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A088868 Numbers n which are divisors of the number formed by concatenating (n-4), (n-3), (n-2) and (n-1) in that order.

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A088870 Numbers n which are divisors of the number produced by concatenating (n-5), (n-4), ... (n-1) in that order.

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