A007908 -id:A007908 - cp's OEIS frontend

A048445 Take the first n numbers written in base 14, concatenate them, then convert from base 14 to base 10.

1, 16, 227, 3182, 44553, 623748, 8732479, 122254714, 1711566005, 23961924080, 335466937131, 4696537119846, 65751519677857, 12887297856859986, 2525910379944557271, 495078434469133225132, 97035373155950112125889, 19018933138566221976674262, 3727710895158979507428155371

Offset: 1

Patrick De Geest, May 15 1999

			a(14) = (1)(2)(3)(4)(5)(6)(7)(8)(9)(A)(B)(C)(D)(10) = 123456789ABCD10_14 = 12887297856859986.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A014897.

Concatenation of first n numbers in other bases: 2: A047778, 3: A048435, 4: A048436, 5: A048437, 6: A048438, 7: A048439, 8: A048440, 9: A048441, 10: A007908, 11: A048442, 12: A048443, 13: A048444, 14: this sequence, 15: A048446, 16: A048447.

[n eq 1 select 1 else Self(n-1)*14^(1+Ilog(14, n))+n: n in [1..20]]; // Vincenzo Librandi, Dec 30 2012

If[STARTPOINT==1, n={}, n=Flatten[IntegerDigits[Range[STARTPOINT-1], 14]]]; Table[AppendTo[n, IntegerDigits[w, 14]]; n=Flatten[n]; FromDigits[n, 14], {w, STARTPOINT, ENDPOINT}] (* Dylan Hamilton, Aug 11 2010 *)
f[n_]:= FromDigits[Flatten@IntegerDigits[Range@n, 14], 14]; Array[f, 20] (* Vincenzo Librandi, Dec 30 2012 *)

A048446 Take the first n numbers written in base 15, concatenate them, then convert from base 15 to base 10.

Original entry on oeis.org

1, 17, 258, 3874, 58115, 871731, 13075972, 196139588, 2942093829, 44131407445, 661971111686, 9929566675302, 148943500129543, 2234152501943159, 502684312937210790, 113103970410872427766, 25448393342446296247367, 5725888502050416655657593, 1288324912961343747522958444

Offset: 1

Patrick De Geest, May 15 1999

			a(14) = (1)(2)(3)(4)(5)(6)(7)(8)(9)(A)(B)(C)(D)(E) = 123456789ABCDE_15 = 2234152501943159.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A014898.

Concatenation of first n numbers in other bases: 2: A047778, 3: A048435, 4: A048436, 5: A048437, 6: A048438, 7: A048439, 8: A048440, 9: A048441, 10: A007908, 11: A048442, 12: A048443, 13: A048444, 14: A048445, 15: this sequence, 16: A048447.

[n eq 1 select 1 else Self(n-1) * 15^(1+Ilog(15, n)) + n: n in [1..20]]; // Vincenzo Librandi, Dec 30 2012

If[STARTPOINT==1, n={}, n=Flatten[IntegerDigits[Range[STARTPOINT-1], 15]]]; Table[AppendTo[n, IntegerDigits[w, 15]]; n=Flatten[n]; FromDigits[n, 15], {w, STARTPOINT, ENDPOINT}] (* Dylan Hamilton, Aug 11 2010 *)
f[n_]:= FromDigits[Flatten@IntegerDigits[Range@n, 15], 15]; Array[f, 20] (* Vincenzo Librandi, Dec 30 2012 *)

A075019 a(1) = 1; for n > 1, a(n) = the smallest prime divisor of the number C(n) formed from the concatenation of 1,2,3,... up to n.

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 127, 2, 3, 2, 3, 2, 113, 2, 3, 2, 3, 2, 13, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 29, 2, 3, 2, 3, 2, 71, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 23, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 10386763, 2, 3, 2, 3, 2, 397, 2, 3, 2, 3, 2, 37907, 2, 3, 2, 3, 2, 73, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 37, 2, 3, 2

Offset: 1

Amarnath Murthy, Sep 01 2002

Least prime factor of A007908(n). For 1 < n <= 5000, a(n) < A007908(n), but this should fail infinitely often (assuming standard heuristics). - Charles R Greathouse IV, Apr 10 2014
From Robert Israel, Aug 28 2015: (Start)
a(n) = 2 iff n is even.
a(n) = 3 iff n == 3 or 5 (mod 6).
a(n) = 5 iff n == 25 (mod 30). (End)

			a(5)= 3, 3 is the smallest prime divisor of 12345.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000 (first 120 terms from Robert Israel)

Cf. A000422, A007908, A075020, A104759, A116504, A116505, A138789, A138790, A138960, A138961, A138962.

C:= 1: A[1]:= 1:
for n from 2 to 100 do
C:= C*10^(1+ilog10(n))+n;
F:= map(t -> t[1],ifactors(C,'easy')[2]);
if hastype(F,integer) then A[n]:= min(select(type,F,integer))
else A[n]:= min(numtheory:-factorset(C))
fi
od:
seq(A[n],n=1..100); # Robert Israel, Aug 28 2015

a = {}; b = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[k]]], {k, Length[w]}]; p = FromDigits[a]; AppendTo[b,First[First[FactorInteger[ p]]]], {n, 25}]; b (* Artur Jasinski, Apr 04 2008 *)

lpf(n)=forprime(p=2,1e3,if(n%p==0,return(p))); factor(n)[1,1]
print1(N=1);for(n=2,100,N=N*10^#Str(n)+n; print1(", "lpf(N))) \\ Charles R Greathouse IV, Apr 10 2014

More terms from Sascha Kurz, Jan 03 2003

A138789 a(n) = number of distinct prime divisors of A104759(n).

Original entry on oeis.org

0, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 4, 4, 2, 5, 4, 2, 3, 7, 4, 4, 3, 3, 2, 5, 3, 5, 2, 3, 6, 6, 3, 3, 5, 5, 7, 3, 3, 4, 3, 3, 4, 4, 5, 3, 5, 7, 3, 5, 6, 6, 5, 4, 5, 2, 6, 4, 6, 4, 4, 7, 6, 5, 5, 6, 9, 5, 5, 7, 5, 5, 5, 5, 6, 6, 4, 4, 4, 5, 8, 7, 6, 4, 5, 4, 4, 9

Offset: 1

Artur Jasinski, Mar 30 2008

Sean A. Irvine, Table of n, a(n) for n = 1..165

Cf. A001221, A104759, A000422, A116504, A007908, A116505.

lst = {}; Do[lst = Join[lst, IntegerDigits[n]], {n, 1, 50}]; Table[Length[FactorInteger[FromDigits[Reverse[lst[[Range[1,n]]]]]]], {n, 1, Length[lst]}] (* Robert Price, Mar 24 2015 *)

a(n) = A001221(A104759(n)). - Michel Marcus, Jun 30 2024

Entire sequence corrected by Robert Price, Mar 24 2015

More terms from Sean A. Irvine, Jul 21 2024

A138793 a(n) = concatenation of reversed digits of natural numbers from n down to 1.

Original entry on oeis.org

1, 21, 321, 4321, 54321, 654321, 7654321, 87654321, 987654321, 1987654321, 1101987654321, 211101987654321, 31211101987654321, 4131211101987654321, 514131211101987654321, 61514131211101987654321, 7161514131211101987654321

Offset: 1

Artur Jasinski, Mar 30 2008, Apr 04 2008

Note that leading zeros are not omitted when writing down digits in reversed order. So 10 reversed becomes 01. - N. J. A. Sloane, Jan 23 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..100

Cf. A104759, A000422, A116504, A007908, A116505, A104759, A138789, A138790.

[Seqint(&cat[Reverse(Intseq(k)): k in [1..n]]): n in [1..16]]; // Bruno Berselli, May 27 2011

read(transforms): A138793 := proc(n) return digrev(parse(cat($(1..n)))): end: seq(A138793(n),n=1..17); # Nathaniel Johnston, Jun 26 2011

b = {}; a = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[k]]], {k, 1, Length[w]}]; p = FromDigits[Reverse[a]]; AppendTo[b, p], {n, 1, 61}]; b (* Artur Jasinski, Mar 30 2008 *)
lst = {}; Table[FromDigits[Reverse[lst = Join[lst, IntegerDigits[n]]]], {n, 1, 15}] (* Robert Price, Mar 22 2015 *)

a(n) = my(s = ""); forstep (k=n,1,-1, sk = digits(k); forstep (j=#sk, 1, -1, s = concat(s, sk[j]))); eval(s); \\ Michel Marcus, Jan 28 2017

A083449 a(n) = A019566(n)/9, where A019566(n) = concat(n,...,1) - concat(1,...,n).

Original entry on oeis.org

0, 1, 22, 343, 4664, 58985, 713306, 8367627, 96021948, -150891621, -13731137410, -260644605199, 86159119727012, 19839246664059223, 3106259112208391434, 422859356777752723645, 53509280234443297055856, 6473262479112108841388067, 759559693477989774385720278

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 01 2003

Are there any palindromes > 58985?

This sequence also gives the number of occurrences of any digit d > n (thus n < 9) in the list of all numbers from 1 to concatenation(1,...,n) = A007908(n) = A014824(n) = sum_{i=1..n} i*10^(n-i). See A277849, A061217, A277830 etc. - M. F. Hasler, Nov 01 2016, edited Nov 07 2020

Cf. A061217.

a:= n-> (parse(cat((n+1-i)$i=1..n))-parse(cat($1..n)))/9:
seq(a(n), n=1..20);  # Alois P. Heinz, Nov 09 2020

Array[(FromDigits@ Apply[Join, Reverse@ #] - FromDigits@ Apply[Join, #])/9 &@ Map[IntegerDigits, Range[#]] &, 19] (* Michael De Vlieger, Nov 12 2020 *)

apply( {A083449(n)=A019566(n)\9}, [1..20]) \\ - M. F. Hasler, Nov 07 2020

For n < 10, a(n) = ceiling((9*n-11)*(10^n+1)/729). - M. F. Hasler, Nov 07 2020

More terms from David Wasserman, Nov 09 2004

A138790 Numbers k such that A138793(k) is prime.

Original entry on oeis.org

61, 946

Offset: 1

Artur Jasinski, Mar 30 2008, Mar 31 2008

There are no more primes for k <= 5000.

a(3) > 20000. - Robert Price, Mar 24 2015

			a(1) = 61 because the number 160695...654321 is prime.

Cf. A000422, A116504, A007908, A116505, A138793.

b = {}; a = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[k]]], {k, 1, Length[w]}]; p = FromDigits[Reverse[a]]; If[PrimeQ[p], Print[n]; AppendTo[b, p]], {n, 1, 2000}]; b (* Artur Jasinski, Mar 30 2008 *)
Select[Range[1, 1000], PrimeQ[lst = {}; Do[lst = Join[lst, IntegerDigits[n]], {n, 1, #}]; FromDigits[Reverse[lst]]] &] (* Robert Price, Mar 24 2015 *)

A262571 Concatenation of the numbers from 2 to n.

Original entry on oeis.org

2, 23, 234, 2345, 23456, 234567, 2345678, 23456789, 2345678910, 234567891011, 23456789101112, 2345678910111213, 234567891011121314, 23456789101112131415, 2345678910111213141516, 234567891011121314151617, 23456789101112131415161718, 2345678910111213141516171819

Offset: 2

N. J. A. Sloane, Sep 25 2015

For primes in this sequence see A089987.
Cf. A007908, A262299, A262572-A262582.
See A262300 for more about this problem.

[Seqint(Reverse(&cat[Reverse(Intseq(k)): k in [2..n]])): n in [2..20]]; // Vincenzo Librandi, Oct 29 2018

Table[FromDigits[Flatten[IntegerDigits[Range[2, n]]]], {n, 2, 19}]  (* Robert Price, Oct 28 2018 *)

def a(n): return int("".join(map(str, range(2, n+1))))
print([a(n) for n in range(2, 20)]) # Michael S. Branicky, Feb 23 2021

A262299 Let S(n) denote the sequence formed by concatenating the decimal numbers 1,2,3,..., omitting n; a(n) is the smallest prime in S(n), or -1 if no term in S(n) is prime.

Original entry on oeis.org

2, 13, 124567, 12356789, 123467891011, 123457, 123456891011

Offset: 1

N. J. A. Sloane and Jerrold B. Tunnell, Sep 25 2015

A262300 is now the main entry for this question.

			a(8) = 1234567910111213...1873 (ending at 1873, a 6384-digit probable prime, and too large to display here) was found by _David Broadhurst_ on Sep 27 2015.
a(9) = 1234567810111213141516171819,
a(11) = 123456789101213,
and a(19) = 12345678910111213141516171820212223242526272829.
Sep 28, 2015: _David Broadhurst_ has also found a(10), a(12), a(14), a(16), a(17), a(18), and a(20). See A262300 for their values.
a(13) is at present unknown.

A262300 gives the last term in S(n) when a prime appears for the first time.
See A262571-A262582 for the sequences S(1) through S(12).
Cf. A007908 (which plays the role of S(0)).
For the primes in S(1) and S(2) see A089987, A262298.

A035333 Concatenation of two or more consecutive positive integers.

Original entry on oeis.org

12, 23, 34, 45, 56, 67, 78, 89, 123, 234, 345, 456, 567, 678, 789, 910, 1011, 1112, 1213, 1234, 1314, 1415, 1516, 1617, 1718, 1819, 1920, 2021, 2122, 2223, 2324, 2345, 2425, 2526, 2627, 2728, 2829, 2930, 3031, 3132, 3233, 3334, 3435, 3456, 3536, 3637, 3738

Offset: 1

Erich Friedman

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

For concatenations of exactly k consecutive integers see A000027 (k=1), A127421 (k=2), A001703 (k=3), A279204 (k=4).
See also A007908 for concatenation of 1 through n.
For primes see A052087.
All of A007908, A052087, A053067, A279610 are subsequences.

import heapq
from itertools import islice
def agen():
    c = 12
    h = [(c, 1, 2)]
    nextcount = 3
    while True:
        (v, s, l) = heapq.heappop(h)
        yield v
        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(), 47))) # Michael S. Branicky, Dec 23 2021

Edited by Charles R Greathouse IV, Apr 28 2010

Corrected by Paul Tek, Jun 08 2013

cp's OEIS Frontend

A048445 Take the first n numbers written in base 14, concatenate them, then convert from base 14 to base 10.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

A048446 Take the first n numbers written in base 15, concatenate them, then convert from base 15 to base 10.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

A075019 a(1) = 1; for n > 1, a(n) = the smallest prime divisor of the number C(n) formed from the concatenation of 1,2,3,... up to n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A138789 a(n) = number of distinct prime divisors of A104759(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A138793 a(n) = concatenation of reversed digits of natural numbers from n down to 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

A083449 a(n) = A019566(n)/9, where A019566(n) = concat(n,...,1) - concat(1,...,n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A138790 Numbers k such that A138793(k) is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A262571 Concatenation of the numbers from 2 to n.

Views

Author