A048436 -id:A048436 - cp's OEIS frontend

A048443 Take the first n numbers written in base 12, concatenate them, then convert from base 12 to base 10.

1, 14, 171, 2056, 24677, 296130, 3553567, 42642812, 511713753, 6140565046, 73686780563, 10610896401084, 1527969081756109, 220027547772879710, 31683966879294678255, 4562491230618433668736, 656998737209054448298001, 94607818158103840554912162

Offset: 1

Patrick De Geest, May 15 1999

			a(12) = (1)(2)(3)(4)(5)(6)(7)(8)(9)(A)(B)(10) = 123456789AB10_12 = 10610896401084.

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

Cf. A014882.

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: this sequence, 13: A048444, 14: A048445, 15: A048446, 16: A048447.

[n eq 1 select 1 else Self(n-1)*12^(1+Ilog(12, n))+n: n in [1..20]]; // Vincenzo Librandi, Dec 30 2012

If[STARTPOINT==1, n={}, n=Flatten[IntegerDigits[Range[STARTPOINT-1], 12]]]; Table[AppendTo[n, IntegerDigits[w, 12]]; n=Flatten[n]; FromDigits[n, 12], {w, STARTPOINT, ENDPOINT}] (* Dylan Hamilton, Aug 11 2010 *)
f[n_]:= FromDigits[Flatten@IntegerDigits[Range@n, 12], 12]; Array[f, 20] (* Vincenzo Librandi, Dec 30 2012 *)

A048444 Take the first n numbers written in base 13, concatenate them, then convert from base 13 to base 10.

Original entry on oeis.org

1, 15, 198, 2578, 33519, 435753, 5664796, 73642356, 957350637, 12445558291, 161792257794, 2103299351334, 355457590375459, 60072332773452585, 10152224238713486880, 1715725896342579282736, 289957676481895898782401, 49002847325440406894225787

Offset: 1

Patrick De Geest, May 15 1999

No primes in the first 31000 terms. - Giovanni Resta, Jun 08 2018

			a(12) = (1)(2)(3)(4)(5)(6)(7)(8)(9)(A)(B)(C) = 123456789ABC_13 = 2103299351334.

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

Cf. A014896.

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: this sequence, 14: A048445, 15: A048446, 16: A048447.

[n eq 1 select 1 else Self(n-1)*13^(1+Ilog(13, n))+n: n in [1..20]]; // Vincenzo Librandi, Dec 30 2012

If[STARTPOINT==1, n={}, n=Flatten[IntegerDigits[Range[STARTPOINT-1], 13]]]; Table[AppendTo[n, IntegerDigits[w, 13]]; n=Flatten[n]; FromDigits[n, 13], {w, STARTPOINT, ENDPOINT}] (* Dylan Hamilton, Aug 11 2010 *)
f[n_]:= FromDigits[Flatten@IntegerDigits[Range@n, 13], 13]; Array[f, 20] (* Vincenzo Librandi, Dec 30 2012 *)

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

Original entry on oeis.org

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 *)

A117640 Concatenation of first n numbers in base 4.

Original entry on oeis.org

1, 12, 123, 12310, 1231011, 123101112, 12310111213, 1231011121320, 123101112132021, 12310111213202122, 1231011121320212223, 123101112132021222330, 12310111213202122233031

Offset: 1

Jonathan Vos Post, Apr 27 2006

Concatenation of the first n terms of A007090.

Base-4 analog of A058935.

N. J. A. Sloane, Table of n, a(n) for n = 1..35

Cf. A030373, A048436.

Other bases: A058935 (2), A360502 (3), A007908 (10).

Table[FromDigits[Flatten[Table[IntegerDigits[n,4],{n,k}]]],{k,15}] (* Harvey P. Dale, Jan 18 2023 *)

from gmpy2 import digits
def A117640(n): return int(''.join(digits(n,4) for n in range(1,n+1))) # Chai Wah Wu, Apr 19 2023

Edited by Jason Kimberley, Nov 27 2012

A350510 Square array read by descending antidiagonals: A(n,k) is the least number m such that the base-n expansion of m contains the base-n expansions of 1..k as substrings.

Original entry on oeis.org

1, 2, 1, 6, 5, 1, 12, 11, 6, 1, 44, 38, 27, 7, 1, 44, 95, 75, 38, 8, 1, 92, 285, 331, 194, 51, 9, 1, 184, 933, 1115, 694, 310, 66, 10, 1, 1208, 2805, 4455, 3819, 1865, 466, 83, 11, 1, 1256, 7179, 17799, 16444, 8345, 3267, 668, 102, 12, 1

Offset: 2

Davis Smith, Jan 02 2022

			Square array begins:
n/k|| 1 |  2 |   3 |    4 |     5 |      6 |       7 |        8 |
================================================================|
2  || 1 |  2 |   6 |   12 |    44 |     44 |      92 |      184 |
3  || 1 |  5 |  11 |   38 |    95 |    285 |     933 |     2805 |
4  || 1 |  6 |  27 |   75 |   331 |   1115 |    4455 |    17799 |
5  || 1 |  7 |  38 |  194 |   694 |   3819 |   16444 |    82169 |
6  || 1 |  8 |  51 |  310 |  1865 |   8345 |   55001 |   289577 |
7  || 1 |  9 |  66 |  466 |  3267 |  22875 |  123717 |   947260 |
8  || 1 | 10 |  83 |  668 |  5349 |  42798 |  342391 |  2177399 |
9  || 1 | 11 | 102 |  922 |  8303 |  74733 |  672604 |  6053444 |
10 || 1 | 12 | 123 | 1234 | 12345 | 123456 | 1234567 | 12345678 |
11 || 1 | 13 | 146 | 1610 | 17715 | 194871 | 2143588 | 23579476 |

Davis Smith, Upper bounds for A350510(n, k) and various conjectured patterns.

Cf. A023811, A035239, A049363, A056744, A057544, A061845, A102305.

The first n - 1 terms of rows: 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: A048446, 16: A048447.

T[n_,k_]:=(m=0;While[!ContainsAll[Subsequences@IntegerDigits[++m,n],IntegerDigits[Range@k,n]]];m);Flatten@Table[T[1+i,j+1-i],{j,9},{i,j}] (* Giorgos Kalogeropoulos, Jan 09 2022 *)

A350510_rows(n,k,N=0)= my(L=List(concat(apply(z->fromdigits([1..z],n),[1..n-1]),if(n>2,fromdigits(concat([1,0],[2..n-1]),n),[]))),T1(x)=digits(x,n),T2(x)=fromdigits(x,n),A(x)=my(S=T1(x));setbinop((y,z)->T2(S[y..z]),[1..#S]),N=if(N,N,L[#L]),A1=A(N));while(#Lsetsearch(A1,z),[1..#L+1])),A1=A(N++));listput(L,N));Vec(L)

For k < n, A(n,k) = A(n,k - 1)*n + k = Sum_{i=1..k} i*(n^(k - i)).
A(n,n) = A049363(n).
A(n,2) = A057544(n).
For n > 3, A(n,3) = A102305(n).
A(n,n - 1) = A023811(n).

A179069 Array read by antidiagonals: row b lists the base-b analog of the base-10 sequence 1, 12, 123, ..., 123456789, 12345678910, ... (A007908).

Original entry on oeis.org

1, 1, 3, 1, 6, 6, 1, 5, 27, 10, 1, 6, 48, 220, 15, 1, 7, 27, 436, 1765, 21, 1, 8, 38, 436, 3939, 14126, 28, 1, 9, 51, 194, 6981, 35367, 113015, 36, 1, 10, 66, 310, 4855, 111702, 318310, 1808248

Offset: 1

Jonathan Vos Post, Jun 27 2010

The numbers in the row b of the array are constructed in base b, but are converted to base 10 for display here.

R. K. Guy writes [UPINT, A3, pp. 9-10]: Selfridge asked if the sequence (in decimal notation) 1, 12, 123, 1234, ... [A007908] ... contains infinitely many primes.... The question can be asked for other scales of notation. There are (trivially) an infinite number of primes in the n=2 column, as that converges to k+2. In the n=3 column, the first prime is A[3,8] = 83 (base 10) = 123 (base 8). In the n=7 column, the first prime is A[8,7] = 342391 (base 10) = 1234567 (base 8). This can be continued to bases higher than 10, where A, B, C, ... are conventionally used as numerals. For example, A[12,5] = 12345 (base 12) = 24677 (base 10) is prime, as is A[12,17] = 656998737209054448298001 (base 10). A[13,3] = 227 (base 10) = 123 (base 13) is prime. Similarly, to pick the 9th row but go further than the table shown here, A[9,14] = 1709671414851143033 (base 10) is prime. Existing OEIS sequences stop at A048447, the concatenation of first n numbers in base 16.

			The array begins:
====================================================================
....|n=1.|.n=2.|.n=3.|.n=4.|..n=5.|..n=6.|...n=7.|.....n=8.|.in OEIS
b=1.|.1..|...3.|...6.|..10.|...15.|...21.|....28.|......36.|.A000217
b=2.|.1..|...6.|..27.|.220.|.1765.|.14126|.113015|.1808248.|.A047778
b=3.|.1..|...5.|..48.|.436.|.3929.|.35367|.318310|.2864798.|.A048435
b=4.|.1..|...6.|..27.|.436.|.6981.|111702|1787239|28595832.|.A048436
b=5.|.1..|...7.|..38.|.194.|.4855.|121381|3034532|75863308.|.A048437
b=6.|.1..|...8.|..51.|.310.|.1865.|.67146|2417263|87021476.|.A048438
b=7.|.1..|...9.|..66.|.466.|.3267.|.22875|1120882|54923226.|.A048439
b=8.|.1..|..10.|..83.|.668.|.5349.|.42798|.342391|21913032.|.A048440
...
b=10|.1..|..12.|.123.|1234.|12345.|123456|1234567|12345678.|.A007908
=====================================================================

Richard K. Guy, Unsolved Problems In Number Theory, 2nd Edn., Springer Verlag, 1994.

Cf. A000217, A007908, A033307, A047778, A048435, A048436, A048437, A048438, A048439, A048440, A048441, A048442, A048443, A048444, A048445, A048446, A048447.

A[b,n] = n-th integer concatenated from consecutive integers in base b.

Should be revised to start with base 2, rather than the ill-defined "base 1". - N. J. A. Sloane, Jul 05 2010

A179075 Concatenation of the first n numbers in base n.

Original entry on oeis.org

6, 48, 436, 4855, 67146, 1120882, 21913032, 490328973, 12345678910, 345227121316, 10610896401084, 355457590375459, 12887297856859986, 502684312937210790, 20988295479420645136, 933876701895122362393, 44111544001370512713990, 2204350295349917301461848

Offset: 2

Jonathan Vos Post, Jun 27 2010

Always divisible by n, hence never prime.

			a(2) = 110 (base 2) = 6 (base 10) = A047778(2).
a(3) = 1210 (base 3) = 48 (base 10) = A048435(3).
a(4) = 12310 (base 4) = 436 (base 10) = A048436(4).
a(5) = 123410 (base 5) = 4855 (base 10) = A048437(5).
a(11) = 123456789A10 (base 11) = 345227121316 (base 10).
a(16) = 123456789ABCDE10 (base 16) = 20988295479420645136 (base 10) = A048447(16).

Cf. A047778, A048435-A048447, A179069.

{for(n=2,19,tlt=0;
for(i=1,n-1,tlt=tlt+i*(n^(n+1-i)));
print1(tlt+n, ", ") )} \\ Douglas Latimer, May 10 2012

a(n) = n + sum(i*(n^(n+1-i)), i=1..n-1).

Edited (errors corrected, sequence extended) by Jon E. Schoenfield, Jul 05 2010 and Jul 06 2010

More terms from Douglas Latimer, May 10 2012

A386985 Smallest k > 0 such that the base-n number formed by concatenating k, k - 1, ..., 2, 1 (each written in base n) is prime, or -1 if no such k exists for the given n.

Original entry on oeis.org

2, 2, 4, 2, 2, 373, 2, 2, 82, 2, 3

Offset: 2

Marco Ripà, Aug 11 2025

If a(13) != -1, then the corresponding prime must have more than 4178 decimal digits.
Sequence continues n=13..36: ?, 2, 2, 28, 362, 2, ?, 2, 2, 4, 2, 3, ?, 2, 5, 4, 2, 2, 37, 3, 2, 4, 2, 2.
The increasing-order analog begins 15, 2, ?. See A048436.

			a(3) = 2 since 21_(base-3) = 7_(base-10), which is prime.

Mathematics Stack Exchange, Concatenation of consecutive positive integers: Does a prime also exist in base 4 (as in bases 2, 3, 5, 6, 7, 8, 9,...)?.
Robert G. Wilson v, Letter to N. J. A. Sloane, Sep. 1992.

Cf. A000422, A048436, A281571.

f(n, b) = my(p=1, L=1); sum(k=1, n, k*p*=L+(k==L&&!L*=b)); \\ adapted from A000422
a(n) = my(k=1); while (!ispseudoprime(f(k, n)), k++); k; \\ Michel Marcus, Aug 16 2025

cp's OEIS Frontend

A048443 Take the first n numbers written in base 12, concatenate them, then convert from base 12 to base 10.

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A048444 Take the first n numbers written in base 13, concatenate them, then convert from base 13 to base 10.

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

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

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A117640 Concatenation of first n numbers in base 4.

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A350510 Square array read by descending antidiagonals: A(n,k) is the least number m such that the base-n expansion of m contains the base-n expansions of 1..k as substrings.

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A179069 Array read by antidiagonals: row b lists the base-b analog of the base-10 sequence 1, 12, 123, ..., 123456789, 12345678910, ... (A007908).

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A179075 Concatenation of the first n numbers in base n.

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