A048435 -id:A048435 - cp's OEIS frontend

A048442 Take the first n numbers written in base 11, concatenate them, then convert from base 11 to base 10.

1, 13, 146, 1610, 17715, 194871, 2143588, 23579476, 259374245, 2853116705, 345227121316, 41772481679248, 5054470283189021, 611590904265871555, 74002499416170458170, 8954302429356625438586, 1083470593952151678068923, 131099941868210353046339701

Offset: 1

Patrick De Geest, May 15 1999

			a(11) = (1)(2)(3)(4)(5)(6)(7)(8)(9)(A)(10) = 123456789A10_11 = 345227121316.

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

Cf. A014881.

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: this sequence, 12: A048443, 13: A048444, 14: A048445, 15: A048446, 16: A048447.

[n eq 1 select 1 else Self(n-1) * 11^(1+Ilog(11, n)) + n: n in [1..20]]; // Vincenzo Librandi, Dec 30 2012

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

{ cuo=0;
for(ixp=1, 18, casi = ixp; cvst=0;
while(casi != 0,
cvd = casi%11; cvst=100*cvst + cvd + 1; casi = (casi - cvd) / 11 );
while(cvst !=0, ptch = cvst%100;
cuo=cuo*11+ptch-1; cvst = (cvst - ptch) / 100 ); print1(cuo, ", "))}
\\ Douglas Latimer, May 09 2012

1 more term from Douglas Latimer, May 10 2012

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

Original entry on oeis.org

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 *)

A360502 Concatenate the ternary strings for 1,2,...,n.

Original entry on oeis.org

1, 12, 1210, 121011, 12101112, 1210111220, 121011122021, 12101112202122, 12101112202122100, 12101112202122100101, 12101112202122100101102, 12101112202122100101102110, 12101112202122100101102110111, 12101112202122100101102110111112, 12101112202122100101102110111112120

Offset: 1

N. J. A. Sloane, Feb 16 2023

If the terms are read as ternary strings and converted to base 10, we get A048435. For example, a(2) = 12_3 = 5_10, which is A048435(2). This is a prime, and gives the first term of A360503.
If the terms are read as decimal numbers, which of them are primes?  12101112202122100101102110111, for example, is not a prime, since it is 37*327057086543840543273030003.
When read as decimal numbers, the first prime is a(7315), with 56003 digits. - Michael S. Branicky, Apr 18 2023

			a(4): concatenate 1, 2, 10, 11, getting 121011.

Winston de Greef, Table of n, a(n) for n = 1..222
Index entries for sequences related to Most Wanted Primes video

Cf. A007089, A036954, A360503, A360504.

This is the ternary analog of A007908.

a:= proc(n) option remember; `if`(n=0, 0, (l-> parse(cat(
      a(n-1), seq(l[-i], i=1..nops(l)))))(convert(n, base, 3)))
    end:
seq(a(n), n=1..15);  # Alois P. Heinz, Feb 17 2023

nn = 15; s = IntegerDigits[Range[nn], 3]; Array[FromDigits[Join @@ s[[1 ;; #]]] &, nn] (* Michael De Vlieger, Apr 19 2023 *)

from sympy.ntheory import digits
def a(n): return int("".join("".join(map(str, digits(k, 3)[1:])) for k in range(1, n+1)))
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Feb 18 2023

# faster version for initial segment of sequence
from sympy.ntheory import digits
from itertools import count, islice
def agen(s=""): yield from (int(s:=s+"".join(map(str, digits(n, 3)[1:]))) for n in count(1))
print(list(islice(agen(), 15))) # Michael S. Branicky, Feb 18 2023

A360506 Read A360505(n) as if it were a base-3 string and write it in base 10.

Original entry on oeis.org

1, 7, 34, 358, 4003, 43369, 456712, 4708240, 47754961, 1339156591, 39693785002, 1169411930926, 34213667699203, 995038950807565, 28790341783585180, 829295063367580492, 23793774263808446005, 680307709052882601259, 19390954850541496025998

Offset: 1

N. J. A. Sloane, Feb 17 2023

This has the same relationship to A360505 as A048435 does to A360502.
The primes in A048435 are in A360503. What are the primes in the present sequence?
Answer: The first primes are a(2) = 7, a(5) = 4003, a(13) = 34213667699203, a(57) and a(109). See A360507. - Rémy Sigrist, Feb 18 2023

			A360505(4) = 111021 and 111021_3 = 358_10 = a(4).

Winston de Greef, Table of n, a(n) for n = 1..407
Index entries for sequences related to Most Wanted Primes video

Cf. A048435, A028898, A360502, A360503, A360505, A360507.

a(n) = fromdigits(concat([digits(k, 3) | k <- Vecrev([1..n])]), 3) \\ Rémy Sigrist, Feb 18 2023

from sympy.ntheory import digits
def a(n): return int("".join("".join(map(str, digits(k, 3)[1:])) for k in range(n, 0, -1)), 3)
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Feb 19 2023

# faster version for initial segment of sequence
from sympy.ntheory import digits
from itertools import count, islice
def agen(s=""): yield from (int(s:="".join(map(str, digits(n, 3)[1:]))+s, 3) for n in count(1))
print(list(islice(agen(), 20))) # Michael S. Branicky, Feb 19 2023

from itertools import count, islice
def A360506_gen(): # generator of terms
    a, b, c = 3, 1, 0
    for i in count(1):
        if i >= a:
            a *= 3
        c += i*b
        yield c
        b *= a
A360506_list = list(islice(A360506_gen(),30)) # Chai Wah Wu, Nov 08 2023

a(n) = A028898(A360505(n)). - Rémy Sigrist, Feb 18 2023

More terms from Rémy Sigrist, Feb 18 2023

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

A360507 Numbers k such that A360506(k) is prime.

Original entry on oeis.org

2, 5, 13, 57, 109, 638, 3069

Offset: 1

Rémy Sigrist and N. J. A. Sloane, Feb 18 2023

Analogous to A360503, which gives the primes in A048435.

			A360506(5) = 4003 is prime, so 5 is a term.

Index entries for sequences related to Most Wanted Primes video

Cf. A048435, A360503, A360506.

a(6)-a(7) from Michael S. Branicky, Feb 18 2023

cp's OEIS Frontend

A048442 Take the first n numbers written in base 11, concatenate them, then convert from base 11 to base 10.

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

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Magma

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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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Magma

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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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Magma

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A360502 Concatenate the ternary strings for 1,2,...,n.

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A360506 Read A360505(n) as if it were a base-3 string and write it in base 10.

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