A048442 -id:A048442 - 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 *)

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

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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Magma

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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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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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