A035333 -id:A035333 - cp's OEIS frontend

A375481 Starting numbers for the terms in A035333.

1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 2, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 3, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 4, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 5, 57, 58

Offset: 1

Nicholas M. R. Frieler, Aug 17 2024

			a(19) = 12 since A035333(19) = 1213, the concatenation of 12 and 13.
a(20) =  1 since A035333(20) = 1234, the concatenation of 1, 2, 3 and 4.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

For the terms generated by consecutive concatenations see A035333.For concatenations of various numbers of consecutive integers see A000027 (k=1), A127421 (k=2), A001703 (k=3), A279204 (k=4).For primes that are the concatenation of two or more consecutive integers see A052087.

ConsecutiveNumber[num_, numberOfNumbers_] := Module[{},
  If[numberOfNumbers == 1, Return[num]];
  FromDigits@(IntegerDigits[num]~Join~
     IntegerDigits@ConsecutiveNumber[num + 1, numberOfNumbers - 1])
 ]
ConsecutiveNumberDigits[maxDigits_] := Module[{numList = {}, c, d},
  Do[
   numList =
     numList~Join~(Association[# -> ConsecutiveNumber[#, c]] & /@
        Range[Power[10, d - 1], Power[10, d] - 1]);,
   {d, 1, Floor[maxDigits/2]}, {c, 2, Floor[maxDigits/d]}
   ];
  SortBy[Select[numList, Values[#][[1]] < Power[10, maxDigits] &],
   Values[#] &]
 ]
Keys[ConsecutiveNumberDigits[8]]//Flatten (* number in this line corresponds to the maximum number of digits of the concatenated terms *)

import heapq
from itertools import islice
def agen(): # generator of terms
    c = 12
    h = [(c, 1, 2)]
    nextcount = 3
    while True:
        (v, s, l) = heapq.heappop(h)
        yield s
        if v >= c:
            c = int(str(c) + str(nextcount))
            heapq.heappush(h, (c, 1, nextcount))
            nextcount += 1
        l += 1; v = int(str(v)[len(str(s)):] + str(l)); s += 1
        heapq.heappush(h, (v, s, l))
print(list(islice(agen(), 70))) # Michael S. Branicky, Aug 18 2024

A053067 a(n) is the concatenation of next n numbers (omit leading 0's).

Original entry on oeis.org

1, 23, 456, 78910, 1112131415, 161718192021, 22232425262728, 2930313233343536, 373839404142434445, 46474849505152535455, 5657585960616263646566, 676869707172737475767778, 79808182838485868788899091, 9293949596979899100101102103104105, 106107108109110111112113114115116117118119120

Offset: 1

Felice Russo, Feb 25 2000

Concatenation of the integers from A000124(n-1) up to and including A000217(n). - R. J. Mathar, Aug 30 2013

The second term is a prime. When is the next prime, if there is another? - N. J. A. Sloane, Dec 16 2016

Felice Russo, A set of new Smarandache functions, sequences and conjectures in number theory, American Research Press 2000.

Michael S. Branicky, Table of n, a(n) for n = 1..200

A subsequence of A035333. For primes in latter, see A052087.

See A279610 for a variant.

Table[FromDigits[Flatten[IntegerDigits/@Range[(n(n-1))/2+1,(n(n+1))/2]]],{n,20}] (* Harvey P. Dale, Jan 23 2016 *)

a(n) = my(s = ""); for (i=n*(n-1)/2 + 1, n*(n+1)/2, s = concat(s, Str(i));); eval(s); \\ Michel Marcus, Aug 11 2017

def a(n): return int("".join(map(str, range((n-1)*n//2+1, n*(n+1)//2+1))))
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Jan 23 2021

More terms from James Sellers, Feb 28 2000

More terms from Michel Marcus, Aug 11 2017

A052087 Primes whose decimal expansion is a concatenation of two or more consecutive increasing numbers (no leading zeros allowed).

Original entry on oeis.org

23, 67, 89, 1213, 3637, 4243, 4567, 5051, 5657, 6263, 6869, 7879, 8081, 9091, 9293, 9697, 102103, 108109, 120121, 126127, 138139, 150151, 156157, 180181, 186187, 188189, 192193, 200201, 216217, 242243, 246247, 252253, 270271

Offset: 1

Patrick De Geest, Jan 15 2000

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A052088, A030458, A052089.

Primes in A035333.

With[{e = 6}, TakeWhile[Select[Function[s, Union@ Flatten@ Map[Function[k, Map[FromDigits@ Flatten@ IntegerDigits@ # &, Partition[Take[s, 10^(e - k)], k, 1]]], Range[2, e]]]@ Range[10^e], PrimeQ], IntegerLength@ # <= e &]] (* Michael De Vlieger, Apr 23 2017 *)

Initial data corrected by Paul Tek, May 07 2013

A001703 Decimal concatenation of n, n+1, and n+2.

Original entry on oeis.org

12, 123, 234, 345, 456, 567, 678, 789, 8910, 91011, 101112, 111213, 121314, 131415, 141516, 151617, 161718, 171819, 181920, 192021, 202122, 212223, 222324, 232425, 242526, 252627, 262728, 272829, 282930, 293031, 303132, 313233, 323334, 333435, 343536, 353637, 363738

Offset: 0

mag(AT)laurel.salles.entpe.fr

All terms are divisible by 3. Every third term starting with a(2) is divisible by 9. - Alonso del Arte, May 27 2013

			a(8) = 8910 since the three consecutive numbers starting with 8 are 8, 9, 10, and these concatenate to 8910. (This is the first term that differs from A193431).

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A074991.
For concatenations of exactly k consecutive integers see A000027 (k=1), A127421 (k=2), A279204 (k=4). For 2 or more see A035333.
See also A127422, A127423, A127424.

read(transforms) :
A001703 := proc(n)
    digcatL([n,n+1,n+2]) ;
end proc:
seq(A001703(n),n=1..20) ; # R. J. Mathar, Mar 29 2017
# Third Maple program:
a:= n-> parse(cat(n, n+1, n+2)):
seq(a(n), n=0..50); # Alois P. Heinz, Mar 29 2017

concat3Nums[n_] := FromDigits@ Flatten@ IntegerDigits[{n, n + 1, n + 2}]; Array[concat3Nums, 25] (* Robert G. Wilson v *)

a(n)=eval(Str(n,n+1,n+2)) \\ Charles R Greathouse IV, Oct 08 2011

for n in range(100): print(int(str(n)+str(n+1)+str(n+2))) # David F. Marrs, Sep 18 2018

The portion of the sequence with all three numbers having d digits - i.e., n in 10^(d-1)..10^d-3 - is in arithmetic sequence: a(n) = (10^(2*d)+10^d+1)*n + (10^d+2). - Franklin T. Adams-Watters, Oct 07 2011

Initial term 12 added and offset changed to 0 at the suggestion of R. J. Mathar. - N. J. A. Sloane, Mar 29 2017

A127421 Numbers whose decimal expansion is a concatenation of 2 consecutive increasing nonnegative numbers.

Original entry on oeis.org

1, 12, 23, 34, 45, 56, 67, 78, 89, 910, 1011, 1112, 1213, 1314, 1415, 1516, 1617, 1718, 1819, 1920, 2021, 2122, 2223, 2324, 2425, 2526, 2627, 2728, 2829, 2930, 3031, 3132, 3233, 3334, 3435, 3536, 3637, 3738, 3839, 3940, 4041, 4142, 4243, 4344, 4445, 4546

Offset: 1

Artur Jasinski, Jan 14 2007

Primes in the sequence are in A030458. - Bruno Berselli, Mar 25 2015

a(n) always has an even number of digits unless n is in A103456. - Alonso del Arte, Oct 30 2019

			a(1) = "0,1" = 1.
a(13) = "12,13" = 1213.

N. J. A. Sloane, Table of n, a(n) for n = 1..25000

A variant of A001704.
For concatenations of exactly k consecutive integers see A000027 (k = 1), A127421 (k = 2), A001703 (k = 3), A279204 (k = 4). For 2 or more see A035333.
Cf. A030458, A127423, A127424, A127425, A103456.

[Seqint(Intseq(n+1) cat Intseq(n)): n in [0..50]]; // Bruno Berselli, Mar 25 2015

a:= n-> parse(cat(n-1, n)):
seq(a(n), n=1..55);  # Alois P. Heinz, Jul 05 2018

nMax = 49; digitsList = IntegerDigits[Range[0, nMax]]; Table[FromDigits[Flatten[{digitsList[[n]], digitsList[[n + 1]]}]], {n, nMax - 1}] (* Alonso del Arte, Oct 24 2019 *)
Table[FromDigits[Flatten[IntegerDigits/@{n,n+1}]],{n,0,50}] (* Harvey P. Dale, May 16 2020 *)

for n in range(100): print(int(str(n)+str(n+1))) # David F. Marrs, Sep 17 2018

val numerStrs = (0 to 49).map(Integer.toString(_)).toList
val concats = (numerStrs.dropRight(1)) zip (numerStrs.drop(1))
concats.map(x => Integer.parseInt(x.1 + x._2)) // _Alonso del Arte, Oct 24 2019

More terms from Joshua Zucker, May 15 2007

A279204 Numbers whose decimal expansion is a concatenation of 4 consecutive increasing nonnegative numbers.

Original entry on oeis.org

1234, 2345, 3456, 4567, 5678, 6789, 78910, 891011, 9101112, 10111213, 11121314, 12131415, 13141516, 14151617, 15161718, 16171819, 17181920, 18192021, 19202122, 20212223, 21222324, 22232425, 23242526, 24252627, 25262728, 26272829, 27282930, 28293031, 29303132, 30313233, 31323334

Offset: 1

N. J. A. Sloane, Dec 17 2016

Primes in this sequence are A030471. Are there infinitely many primes in the sequence? - Chai Wah Wu, Dec 17 2016

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A030471 (primes).

For concatenations of exactly k consecutive integers see A000027(k=1), A127421 (k=2), A001703 (k=3), A279204 (k=4). For 2 or more see A035333.

A279204[n_] := FromDigits[Flatten[IntegerDigits[Range[n, n + 3]]]];
Array[A279204, 50] (* Paolo Xausa, Aug 26 2024 *)

def A279204(n):
    return int(str(n)+str(n+1)+str(n+2)+str(n+3)) # Chai Wah Wu, Dec 17 2016

A094333 a(1) = 1, a(n) = least proper multiple of a(n-1) which is a concatenation of successive numbers. Terms with more than one digit must be a concatenation of at least two successive numbers.

Original entry on oeis.org

1, 2, 4, 8, 56, 5152, 17181920, 479839479840, 552587148319552587148320, 370210294386175240152319370210294386175240152320

Offset: 1

Amarnath Murthy, May 16 2004

			a(3) = 4 is the smallest proper multiple of 2 which is a concatenation of a (single) number.
a(5) = 56 and a(6) = 5152 =56*92 is the concatenation of 51 and 52.

Paul Tek, Table of n, a(n) for n = 1..14

Cf. A035333, A094334, A225796.

a(7), a(8) from D. S. McNeil, Mar 23 2009
a(9) from Donovan Johnson, Mar 09 2011
a(10) from Paul Tek, Jul 27 2013

A279610 a(n) = concatenate n consecutive integers, starting with the last number of the previous batch.

Original entry on oeis.org

1, 12, 234, 4567, 7891011, 111213141516, 16171819202122, 2223242526272829, 293031323334353637, 37383940414243444546, 4647484950515253545556, 565758596061626364656667, 67686970717273747576777879, 7980818283848586878889909192

Offset: 1

José de Jesús Camacho Medina, Dec 09 2016, and Paolo Iachia, Dec 15 2016

A variant of A053067. The first number of the concatenation a(n) is A152947(n) = (n-2)*(n-1)/2+1 and the last is (n-1)*n/2+1.
The fourth term, 4567, is a prime. When is the next prime, if there is another? - N. J. A. Sloane, Dec 16 2016
a(n) is the concatenation of the terms of the n-th row of A122797 when seen as a triangle. - Michel Marcus, Dec 17 2016

			a(4) is the concatenation of 4 numbers beginning with the last number (4) that was used to build a(3), so a(4) = 4 5 6 7 = 4567. Then a(5) is the concatenation of 5 numbers beginning with the last number of a(4), which is 7, so a(5) = 7 8 9 10 11 = 7891011. And so on.
For n = 3, n^2/2 - n/2 + 1 = 4; a(3) = 4 + 3*10^1 + 2*10^(1+1) = 234.

Chai Wah Wu, Table of n, a(n) for n = 1..200

Cf. A053067, A122797, A152947.

A subsequence of A035333. For primes in latter, see A052087.

Table[FromDigits[Flatten[IntegerDigits /@ Range[(n(n - 1))/2 + 1, (n(n + 1))/2 + 1 ]]], {n, 0, 20}]

from _future_ import division
def A279610(n):
    return int(''.join(str(d) for d in range((n-1)*(n-2)//2+1,n*(n-1)//2+2))) # Chai Wah Wu, Dec 17 2016

a(n) = n^2/2 - n/2 + 1 + Sum{k=1..n-1} ((n^2/2 - n/2 + 1 - k)*10^Sum{j=0..k-1} (floor(1+log_10(n^2/2 - n/2 + 1 - j)))).

A375461 Self-consecutive numbers: numbers m such that there exists a nonnegative integer k for which the concatenation of m, m+1, ..., m+k is an m-digit number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168

Offset: 1

Nicholas M. R. Frieler, Aug 17 2024

Let d be the number of digits of m (in base 10). One can show that the number m is a member of this sequence if and only if:
    1) m is divisible by d AND m < 10^d * d/(d+1)
     OR
    2) 10^d is divisible by d+1 AND m >= 10^d * d/(d+1).
Equivalently, integers m in the block 10^(d-1) <= m < 10^d are included in this sequence based on their relation to the value s = 10^d * d/(d+1). If m < s, then m is in this sequence iff m is divisible by d. If m >= s, m is in this sequence iff s is an integer. Note that for d = 2, s = 66.666... so no integers m in the range 66.666... <= m < 100 are in this sequence. For d = 3, s = 750 is an integer so all integers m in the range 750 <= m < 1000 are in this sequence. - Dominic McCarty, Sep 09 2024

			7 is a term because if we concatenate 7, 8, 9, 10, and 11, we get 7891011, which has 7 digits.
102 is a term because we can concatenate 102 with the next 33 consecutive integers to get a number with 34*3 = 102 digits.
68 is not a term because if we concatenate 68,69,...,100 we get a 67-digit number and concatenating 101 with this yields a 70-digit number, so it is not possible to achieve a 68-digit number.

Dominic McCarty, Table of n, a(n) for n = 1..10000

Cf. A035333 (concatenations), A375590.

function next(m){let d=(""+m).length;if(m*(d+1)d*(10**d)){return next(10**d)}else{return m+d}}else{if((10**d)%(d+1)==0){return m+1}else{return next(10**d)}}}let m=0;a="";while(m<168){m=next(m);a+=m+", "}console.log(a) // Dominic McCarty, Sep 09 2024

SelfConsecutiveQ[n_] :=
 Module[{len = Length@IntegerDigits[n], counter = 1, numDigits = 0},
  If[len*(Power[10, len] - n) >= n, Return[Mod[n, len] == 0],
   numDigits = len*(Power[10, len] - n)];
  Mod[n - numDigits, len + 1] == 0
 ]
Select[Range[1000], SelfConsecutiveQ[#] &]

A225796 a(1) = 1, a(n) = least proper multiple of a(n-1) which is a concatenation of two or more successive numbers.

Original entry on oeis.org

1, 12, 456, 415416, 194191194192, 3816727961538167279616, 35270249819024056401913527024981902405640192

Offset: 1

Paul Tek, Jul 27 2013

			12 is the least proper multiple of 1 in A035333, hence a(2)=12.
456 is the least proper multiple of 12 in A035333, hence a(3)=456.

Paul Tek, Table of n, a(n) for n = 1..11
Paul Tek, PARI program for this sequence

Cf. A035333, A094333.

PARI
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See Links section.
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cp's OEIS Frontend

A375481 Starting numbers for the terms in A035333.

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A053067 a(n) is the concatenation of next n numbers (omit leading 0's).

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A052087 Primes whose decimal expansion is a concatenation of two or more consecutive increasing numbers (no leading zeros allowed).

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A001703 Decimal concatenation of n, n+1, and n+2.

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A127421 Numbers whose decimal expansion is a concatenation of 2 consecutive increasing nonnegative numbers.

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Magma

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A279204 Numbers whose decimal expansion is a concatenation of 4 consecutive increasing nonnegative numbers.

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A094333 a(1) = 1, a(n) = least proper multiple of a(n-1) which is a concatenation of successive numbers. Terms with more than one digit must be a concatenation of at least two successive numbers.

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