A020338 -id:A020338 - cp's OEIS frontend

A324262 a(n) is the smallest term of A324261 with 2n digits if it exists, otherwise 0.

0, 0, 0, 0, 13731373, 1190911909, 0, 19090911909091, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7316763215964848081373167632159648480813, 111272689909091345969111272689909091345969, 10527889691056689261011052788969105668926101, 0, 0, 0, 0

Offset: 0

Deron Stewart, Mar 13 2019

There can be a large number of terms in A324261 with 2n digits. For example, there are 227 terms with length 66. Selecting only the smallest term for each length allows terms to be listed for larger values of n.
The terms of A324261 with length 2n are formed by concatenating two copies of prime p, where p has length n and the decimal representation of p contains all the prime factors of 10^n + 1 as described in A324261 and A083359.
Subsequence of A020338 (Doublets: base-10 representation is the juxtaposition of two identical strings).

			With m = 27 there are three prime p's: 114175966169705419295257913, 352579141759661697054192911 and 525791141759661697054192913. The smallest p concatenated with itself gives a(27) = 114175966169705419295257913114175966169705419295257913.
With m = 28 there are no solutions so a(28) = 0.

Deron Stewart, Table of n, a(n) for n = 0..61

Cf. A083359, A020338, A324261, A324257.

A345391 a(n) is the least proper multiple of n with the same set of decimal digits as n.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 100, 1111, 1212, 1131, 1414, 1155, 1616, 1717, 1188, 1919, 200, 2121, 2222, 322, 2424, 225, 2262, 2727, 2828, 2929, 300, 1333, 3232, 3333, 3434, 3535, 3636, 3737, 3838, 3393, 400, 4141, 4242, 344, 4444, 4455, 644, 4747

Offset: 1

Rémy Sigrist, Jun 17 2021

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A020338, A029793, A276769.

a(n) = { my (d=Set(digits(n))); forstep (m=2*n, oo, n, if (Set(digits(m))==d, return (m))) }

def a(n):
    kn, ss = 2*n, set(str(n))
    while set(str(kn)) != ss: kn += n
    return kn
print([a(n) for n in range(1, 49)]) # Michael S. Branicky, Jun 17 2021

a(n) <= A020338(n) for any n > 0.

A385623 Array read by ascending antidiagonals: A(n,k) is the number obtained by concatenation of n with k in that order, with k >= 0.

Original entry on oeis.org

0, 10, 1, 20, 11, 2, 30, 21, 12, 3, 40, 31, 22, 13, 4, 50, 41, 32, 23, 14, 5, 60, 51, 42, 33, 24, 15, 6, 70, 61, 52, 43, 34, 25, 16, 7, 80, 71, 62, 53, 44, 35, 26, 17, 8, 90, 81, 72, 63, 54, 45, 36, 27, 18, 9, 100, 91, 82, 73, 64, 55, 46, 37, 28, 19, 10, 110, 101, 92, 83, 74, 65, 56, 47, 38, 29, 110, 11

Offset: 0

Stefano Spezia, Jul 05 2025

			Array begins as:
   0,  1,  2,  3,  4,  5,  6,  7, ...
  10, 11, 12, 13, 14, 15, 16, 17, ...
  20, 21, 22, 23, 24, 25, 26, 27, ...
  30, 31, 32, 33, 34, 35, 36, 37, ...
  40, 41, 42, 43, 44, 45, 46, 47, ...
  50, 51, 52, 53, 54, 55, 56, 57, ...
  60, 61, 62, 63, 64, 65, 66, 67, ...
  ...

Stefano Spezia, Table of n, a(n) for n = 0..11475 (first 151 antidiagonals of the array, flattened)

Columns for k=0..9 give: A008592, A017281, A017293, A017305, A017317, A017329, A017341, A017353, A017365, A017377.

Cf. A001477 (1st row), A020338 (main diagonal), A055642, A385624 (antidiagonal sums).

A[n_,k_]:=FromDigits[Join[IntegerDigits[n],IntegerDigits[k]]]; Table[A[n,k],{n,0,6},{k,0,7}] (* or *)
A[n_,k_]:=If[k==0,10n,n*10^(Floor[Log10[k]]+1)+k]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

T(n, k) = fromdigits(concat(digits(n), digits(k))); \\ Michel Marcus, Jul 06 2025

A(n,0) = 10*n and A(n,k) = n*10^(floor(log_10(k)) + 1) + k for k > 0.

A074843 Quadruplets: base 10 representation is the juxtaposition of four identical strings.

Original entry on oeis.org

1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999, 10101010, 11111111, 12121212, 13131313, 14141414, 15151515, 16161616, 17171717, 18181818, 19191919, 20202020, 21212121, 22222222, 23232323, 24242424, 25252525

Offset: 1

Felice Russo, Sep 10 2002

Doublets in which the index is also a doublet. - Jamie Robert Creasey, Jun 23 2021

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A020338.

Table[FromDigits[Flatten[IntegerDigits/@PadRight[{},4,n]]],{n,30}] (* Harvey P. Dale, Dec 20 2021 *)

a(n)=eval(Str(n,n,n,n)) \\ Charles R Greathouse IV, Jun 23 2021

10 cls 30 for I=1 to 100 40 A=str(I) 50 C=A+A+A+A 60 B=val(cutspc(C)) 80 print B 90 next 100 end

A020338(A020338(n)). - Jamie Robert Creasey, Jun 23 2021

A105814 a(n) = n^2 + (n concatenated with n).

Original entry on oeis.org

12, 26, 42, 60, 80, 102, 126, 152, 180, 1110, 1232, 1356, 1482, 1610, 1740, 1872, 2006, 2142, 2280, 2420, 2562, 2706, 2852, 3000, 3150, 3302, 3456, 3612, 3770, 3930, 4092, 4256, 4422, 4590, 4760, 4932, 5106, 5282, 5460, 5640, 5822, 6006, 6192, 6380, 6570

Offset: 1

Eric Angelini, May 05 2005

			1 + 11 = 12, 4 + 22 = 26, 9 + 33 = 42, 16 + 44 = 60, ..., 100 + 1010 = 1110.

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

Cf. A020338.

[Seqint(Intseq(n) cat Intseq(n))+n^2: n in [1..50]]; // Vincenzo Librandi, Jul 31 2015

[n^2+n*(1+10^(1+Floor(Log(n)/Log(10)))): n in [1..50]]; // Vincenzo Librandi, Jul 31 2015

seq(n^2 + n*(1+10^(1+ilog10(n))),n=1..100); # Robert Israel, Jul 30 2015

f[n_] := Block[{id = IntegerDigits[n]}, n^2 + FromDigits[ Join[id, id]]]; Table[ f[n], {n, 45}] (* Robert G. Wilson v, May 10 2005 *)

vector(50, n, n^2 + eval(Str(n,n))) \\ Michel Marcus, Jul 31 2015

def a(n): return n**2 + int(str(n)*2)
print([a(n) for n in range(1, 47)]) # Michael S. Branicky, Dec 26 2021

a(n) = n^2 + n*(1+10^(1+floor(log_10(n)))). - Robert Israel, Jul 30 2015

a(n) = n^2 + A020338(n). - Michel Marcus, Jul 31 2015

More terms from Robert G. Wilson v, May 10 2005

A206529 7^n concatenated with itself.

Original entry on oeis.org

11, 77, 4949, 343343, 24012401, 1680716807, 117649117649, 823543823543, 57648015764801, 4035360740353607, 282475249282475249, 19773267431977326743, 1384128720113841287201, 9688901040796889010407, 678223072849678223072849

Offset: 0

Vincenzo Librandi, Mar 18 2012

			a(1)=77 because 7^1 concatenated with 7^1 is 77.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000420, A000351, A178664, A206527, A206528.

[Seqint(Intseq(7^n) cat Intseq(7^n)): n in [0..20]];

Table[FromDigits[Join[IntegerDigits[7^n],IntegerDigits[7^n]]],{n,0,20}]
#*10^IntegerLength[#]+#&/@(7^Range[0,20]) (* Harvey P. Dale, Jul 19 2022 *)

a(n) = A020338(7^n).

A222962 Primes of the form kk*k+k+1, where kk is the concatenation of k with itself.

Original entry on oeis.org

13, 47, 103, 181, 281, 547, 10111, 14557, 22741, 25873, 29207, 44563, 48907, 53453, 90931, 103457, 110023, 116791, 161641, 169823, 178207, 186793, 195581, 232753, 242551, 273157, 283763, 305581, 316793, 440023, 523657, 538303, 568201, 614563, 662743

Offset: 1

Vincenzo Librandi, Mar 18 2013

Corresponding values of k are: 1, 2, 3, 4, 5, 7, 10, 12, 15, 16, 17, 21, 22, 23, 30, 32, 33, 34, 40, 41, 42, 43, 44, 48, 49, 52, 53, 55, 56, 66, 72, 73, 75, 78, 81, 82, 83, 92,...

a(7), a(43) and a(204) (see b-file) have the form 10^(3n+1)+10^(2n)+10^n+1 = (10^(n+1)*10^n+10^n)*10^n+10^n+1. The next term of this type is 10^247+10^164+10^82+1.

			22741 is in the sequence because it is prime and 22741=1515*15+15+1.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A020338, A157711.

[p: n in [0..100] | IsPrime(p) where p is Seqint(Intseq(n) cat Intseq(n))*n+n+1]; // Bruno Berselli, Mar 21 2013

f[n_] := FromDigits@Flatten@IntegerDigits[{n, n}] n + n + 1; Select[Table[f[n], {n, 100}], PrimeQ] (* Bruno Berselli, Mar 21 2013 *)
Select[Table[n(n*10^IntegerLength[n]+n)+n+1,{n,100}],PrimeQ] (* Harvey P. Dale, Oct 29 2023 *)

Edited by Bruno Berselli, Mar 22 2013

A248126 a(n) = n^2 with each digit repeated.

Original entry on oeis.org

11, 44, 99, 1166, 2255, 3366, 4499, 6644, 8811, 110000, 112211, 114444, 116699, 119966, 222255, 225566, 228899, 332244, 336611, 440000, 444411, 448844, 552299, 557766, 662255, 667766, 772299, 778844, 884411, 990000, 996611, 11002244, 11008899, 11115566

Offset: 1

Jon Perry, Nov 01 2014

Inspired by A116699.

			13^2 = 169, so a(13) = 116699.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A020338, A044836.

Cf. A116699, A338754.

for (i=1;i<40;i++) {
s=(i*i).toString();
for (j=0;j

a248126[n_Integer] :=
Module[{m}, m := IntegerDigits[n^2];
  FromDigits[Flatten[Transpose[List[m, m]]]]]; a248126 /@ Range[34] (* Michael De Vlieger, Nov 06 2014 *)
Table[FromDigits[Riffle[id=IntegerDigits[n^2],id]],{n,40}] (* Harvey P. Dale, Dec 19 2015 *)

a(n)= my(d = 11*digits(n^2)); fromdigits(d, 100) \\ David A. Corneth, Dec 02 2023

def a(n): return int("".join(d*2 for d in str(n**2)))
print([a(n) for n in range(1, 35)]) # Michael S. Branicky, Dec 02 2023

A248365 4n concatenated with itself.

Original entry on oeis.org

44, 88, 1212, 1616, 2020, 2424, 2828, 3232, 3636, 4040, 4444, 4848, 5252, 5656, 6060, 6464, 6868, 7272, 7676, 8080, 8484, 8888, 9292, 9696, 100100, 104104, 108108, 112112, 116116, 120120, 124124, 128128, 132132, 136136, 140140, 144144, 148148, 152152, 156156

Offset: 1

Jacy Fang, Oct 05 2014

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A020338 (n), A248038 (3n), A248422 (2n).

a248038[n_Integer] := Module[{m, t}, m := ToString[4 n ]; ToExpression[m <> m]]; a248038 /@ Range[50]; (* Michael De Vlieger, Nov 06 2014 *)

{print(str(4*n)*2,end=', ') for n in range(1,50)} # Derek Orr, Oct 05 2014

A324261 Subsequence of A083359 (Visual Factor Numbers) of the form (10^m+1)*p, where the decimal representation of prime p contains all the prime factors of 10^m+1.

Original entry on oeis.org

13731373, 31373137, 1190911909, 9091190911, 19090911909091, 7316763215964848081373167632159648480813, 111272689909091345969111272689909091345969, 111279090913268945969111279090913268945969, 112726894596919090913112726894596919090913

Offset: 1

Deron Stewart, Mar 13 2019

Subsequence of A020338 (Doublets: base-10 representation is the juxtaposition of two identical strings).
The prime factors of 10^m + 1 are also prime factors of 10^km + 1, where k is odd, so a(n) and a(km) have those prime factors in common.
Expanding on the comment in A083359 regarding finding large terms, we can generate large doublet terms in the following way:
-- for any composite number M = 10^m + 1 try to compose a prime number p of length m from the prime factors of M. Generally this will require the factors to overlap to reduce the length to m.
-- the factors can wrap around to the beginning of p. For example, if M has a factor of 137 then p can be of the form 7...13.
-- the term is formed by concatenating p with itself to form a(n) = p||p. The resulting number will consist entirely of the concatenation of its prime factors with allowed overlap as in A083359.

			With m = 4: 10^4 + 1 = 10001 = 73 * 137. We can form prime p = 1373 which concatenates with itself to give a(1) = 13731373 = 73 * 137 * 1373. We can also form the prime p = 3137 which gives a(2). The number 7313 also contains all the prime factors of 10001 but it is not prime.
With m = 33: 10^33 + 1 = 7*11*11*13*23*4093*8779*599144041*183411838171, there are 4932 m-digit numbers that contain all the factors, of which 227 of them are prime. Each of these primes generates a term in the sequence with 66 digits, the smallest of which is 112359914404134093877918341183817112359914404134093877918341183817. This is A324262(33).

Cf. A083359, A020338, A324262, A324257.

cp's OEIS Frontend

A324262 a(n) is the smallest term of A324261 with 2n digits if it exists, otherwise 0.

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A345391 a(n) is the least proper multiple of n with the same set of decimal digits as n.

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A385623 Array read by ascending antidiagonals: A(n,k) is the number obtained by concatenation of n with k in that order, with k >= 0.

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A074843 Quadruplets: base 10 representation is the juxtaposition of four identical strings.

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A105814 a(n) = n^2 + (n concatenated with n).

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A206529 7^n concatenated with itself.

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A222962 Primes of the form kk*k+k+1, where kk is the concatenation of k with itself.

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