A048376 -id:A048376 - cp's OEIS frontend

A057692 Smallest prime which produces exactly n+1 different primes after n applications of the A048376 transform.

2, 31, 641, 12422153, 66132153133

Offset: 0

G. L. Honaker, Jr., Oct 20 2000

a(4) found by Carlos Rivera and confirmed to be the smallest by Paul Jobling (Paul.Jobling(AT)WhiteCross.com)

a(5) = 66132153133 leads to a final (probable) prime of 3560 digits. If zero is allowed, then a(5) = 12505785661 and the last (probable) prime would have 10982 digits. - Giovanni Resta, Sep 15 2011

			31 becomes 3331 and both 31 and 3331 are primes. 641 becomes 66666644441 and then 66666666666666666666666666666666666644444444444444441 and all 3 are primes.

C. Rivera (Ed.), Puzzle 112. Automorphic primes, primepuzzles.net. (Published Oct. 2000 or earlier.)

a(1,2,3,...) is a subsequence of A057628.

A057692(n,s=2)={ forprime(p=s,, my(q=p); for(i=2,n, isprime(q=A048376(q))||next(2)); isprime(A048376(q))||return(p))} \\ Impractical for n>3. - M. F. Hasler, Jan 23 2013

a(5) from Giovanni Resta, Sep 15 2011

Definition corrected by M. F. Hasler, Jan 23 2013

A000461 Concatenate n n times.

Original entry on oeis.org

1, 22, 333, 4444, 55555, 666666, 7777777, 88888888, 999999999, 10101010101010101010, 1111111111111111111111, 121212121212121212121212, 13131313131313131313131313, 1414141414141414141414141414, 151515151515151515151515151515, 16161616161616161616161616161616

Offset: 1

John Radu (Suttones(AT)aol.com)

			From _Bruno Berselli_, Oct 05 2018: (Start)
.         1 * 9 = 09
.        22 * 9 = 198
.       333 * 9 = 2997
.      4444 * 9 = 39996
.     55555 * 9 = 499995
.    666666 * 9 = 5999994
.   7777777 * 9 = 69999993
.  88888888 * 9 = 799999992
. 999999999 * 9 = 8999999991
(End)

F. Smarandache, "Properties of the numbers", Univ. of Craiova Archives, 1975; Arizona State University Special Collections, Tempe, AZ.

Reinhard Zumkeller, Table of n, a(n) for n = 1..333
Eric Weisstein's World of Mathematics, Smarandache Sequences

Cf. A048376, A053422.

a000461 n = (read $ concat $ replicate n $ show n) :: Integer
-- Reinhard Zumkeller, Apr 26 2011

a:= n-> parse(cat(n$n)):
seq(a(n), n=1..20);  # Alois P. Heinz, Apr 26 2011

Table[Sum[(n)*10^(i*(Floor[Log[10, n]] + 1)), {i, 0, n - 1}], {n, 1, 30}] (* José de Jesús Camacho Medina, Dec 10 2014 *)
Table[FromDigits[Flatten[IntegerDigits/@Table[n,{n}]]],{n,15}] (* Harvey P. Dale, Mar 01 2015 *)
Table[FromDigits[PadRight[{},n IntegerLength[n],IntegerDigits[n]]],{n,15}] (* Harvey P. Dale, Jun 19 2016 *)

a(n) = eval(concat(apply(x->Str(x), vector(n, k, n)))); \\ Michel Marcus, Oct 05 2018; Feb 12 2023

def a(n): return int(str(n)*n)
print([a(n) for n in range(1, 17)]) # Michael S. Branicky, Jan 22 2021

a(n) = n*(10^(n*L(n))-1)/(10^L(n)-1) where L(n) = A004216(n)+1 = floor(log_10(10n)). - Henry Bottomley, Jun 01 2000
A055642(a(n)) = n * A055642(n). - Reinhard Zumkeller, Apr 26 2011
a(n) = Sum_{i=0..n-1} (n*10^(i*(floor(log(10, n)) + 1))). - José de Jesús Camacho Medina, Dec 10 2014

A053422 n times (n 1's): a(n) = n*(10^n - 1)/9.

Original entry on oeis.org

0, 1, 22, 333, 4444, 55555, 666666, 7777777, 88888888, 999999999, 11111111110, 122222222221, 1333333333332, 14444444444443, 155555555555554, 1666666666666665, 17777777777777776, 188888888888888887, 1999999999999999998, 21111111111111111109, 222222222222222222220, 2333333333333333333331

Offset: 0

Henry Bottomley, Mar 07 2000

R_a(n) is the least repunit divisible by the square of R_n = (10^n - 1)/9. - Lekraj Beedassy, Jun 07 2006

G. C. Greubel, Table of n, a(n) for n = 0..995
Index entries for linear recurrences with constant coefficients, signature (22,-141,220,-100).

Cf. A000461, A002275, A048376.

I:=[0, 1, 22, 333]; [n le 4 select I[n] else 22*Self(n-1) - 141*Self(n-2) +220*Self(n-3) -100*Self(n-4): n in [1..30]]; // G. C. Greubel, May 25 2018

LinearRecurrence[{22,-141,220,-100}, {0, 1, 22, 333}, 50] (* G. C. Greubel, May 25 2018 *)
CoefficientList[Series[x (1-10x^2)/((1-x)^2(1-10x)^2),{x,0,30}],x] (* Harvey P. Dale, Jun 29 2021 *)

x='x+O('x^30); concat([0], Vec(x*(1-10*x^2)/((1-x)^2*(1-10*x)^2))) \\ G. C. Greubel, May 25 2018

[gaussian_binomial(n,1,10)*n for n in range(0,22)] # Zerinvary Lajos, May 29 2009

a(n) = n*A002275(n) = a(n-1)*10n/(n-1) + n.
O.g.f.: x*(1-10*x^2)/((1-x)^2*(1-10*x)^2). - R. J. Mathar, Jan 21 2008
E.g.f.: x*exp(x)*(10*exp(9*x) - 1)/9. - Stefano Spezia, Sep 14 2023

Corrected by Jason Earls, Sep 02 2006

A048377 Append d digits d after each digit d in decimal expansion of n.

Original entry on oeis.org

0, 11, 222, 3333, 44444, 555555, 6666666, 77777777, 888888888, 9999999999, 110, 1111, 11222, 113333, 1144444, 11555555, 116666666, 1177777777, 11888888888, 119999999999, 2220, 22211, 222222, 2223333, 22244444, 222555555, 2226666666

Offset: 0

Patrick De Geest, Mar 15 1999

a(10^n) = 11 * 10^n; A007953(a(n)) = A003132(n) + A007953(n). [Reinhard Zumkeller, Jul 10 2011]

			12 becomes 1 1 2 22 = 11222.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A048376.

import Data.Char (digitToInt)
a048377 :: Integer -> Integer
a048377 n =
   read $ concat $ zipWith replicate (map ((+ 1) . digitToInt) ns) ns
      where ns = show n
-- Reinhard Zumkeller, Jul 10 2011

den[n_]:=Module[{idn=IntegerDigits[n]},FromDigits[Flatten[Table[ Table[ idn[[i]],{idn[[i]]+1}],{i,Length[idn]}]]]]; Array[den,30,0] (* Harvey P. Dale, Sep 04 2013 *)

A057628 Primes such that replacing each digit d with d copies of the digit d produces a prime. Zeros are not allowed.

Original entry on oeis.org

11, 31, 53, 131, 149, 223, 283, 311, 313, 331, 397, 463, 641, 691, 937, 941, 1439, 1511, 1741, 1871, 1949, 1993, 1999, 2111, 2447, 2939, 3163, 3391, 3433, 3499, 3559, 3593, 3659, 3911, 3931, 5227, 5399, 5923, 6163, 6269, 6653, 6719, 7177, 7741, 8389

Offset: 1

G. L. Honaker, Jr., Oct 10 2000

"Replacing each digit d with d copies of the digit d" is the function A048376. Therefore this is the largest subset of A038618 stable under the map A048376.

			E.g. 641 becomes 66666644441 which is also prime.

Cf. A038618, A048376, A057630.

Select[Prime[Range[1500]],PrimeQ[FromDigits[Flatten[Table[#,{#}]&/@ IntegerDigits[#]]]]&&DigitCount[#,10,0]==0&]  (* Harvey P. Dale, Mar 27 2011 *)

is_A057628(n)={vecmin(digits(n)) && is_A057630(n)} \\ M. F. Hasler, Jan 23 2013

More terms from Patrick De Geest, Oct 15 2000.

Offset changed to 1, according to OEIS conventions, by M. F. Hasler, Jan 23 2013

A057630 Primes such that replacing each digit d with d copies of the digit d produces a prime. Zeros are allowed.

Original entry on oeis.org

11, 31, 53, 101, 131, 149, 223, 283, 311, 313, 331, 397, 463, 503, 641, 691, 937, 941, 1031, 1049, 1069, 1301, 1409, 1439, 1511, 1609, 1741, 1871, 1949, 1993, 1999, 2083, 2111, 2203, 2447, 2803, 2939, 3001, 3011, 3061, 3163, 3301, 3391, 3433, 3499, 3559

Offset: 1

G. L. Honaker, Jr., Oct 10 2000

"Replacing each digit d with d copies of the digit d" is the function A048376, well defined on the set of positive integers. Therefore (the range of) the present sequence is the largest subset of A000040 stable under the operation A048376.

A004022 is a subsequence. - Chai Wah Wu, Dec 19 2019

			E.g. 641 becomes 66666644441 which is also prime.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A004022, A057628.

Select[Prime[Range[500]],PrimeQ[FromDigits[Flatten[Table[#,{#}]&/@ IntegerDigits[ #]]]]&]  (* Harvey P. Dale, Dec 18 2010 *)

is_A057630(n)={isprime(A048376(n)) && isprime(n)} \\ M. F. Hasler, Jan 23 2013

from sympy import isprime, nextprime
A057630_list, dlist, p = [], [str(d)*d for d in range(10)], 2
while len(A057630_list) < 10000:
    if isprime(int(''.join(dlist[int(d)] for d in str(p)))):
        A057630_list.append(p)
    p = nextprime(p) # Chai Wah Wu, Dec 19 2019, corrected Jan 01 2022

More terms from Patrick De Geest, Oct 15 2000

Offset changed to 1, according to OEIS conventions, by M. F. Hasler, Jan 23 2013

A158849 a(10n+m) is the integer with n+m concatenations of the digit m in base 10, 0<=m<=9.

Original entry on oeis.org

1, 22, 333, 4444, 55555, 666666, 7777777, 88888888, 999999999, 0, 11, 222, 3333, 44444, 555555, 6666666, 77777777, 888888888, 9999999999, 0, 111, 2222, 33333, 444444, 5555555, 66666666, 777777777, 8888888888, 99999999999, 0, 1111

Offset: 1

Paul Curtz, Mar 28 2009

A000461, A048376.

Edited by R. J. Mathar, Apr 04 2009

A118115 Partial sums of n concatenated n times.

Original entry on oeis.org

1, 23, 356, 4800, 60355, 727021, 8504798, 97393686, 1097393685, 10101010102107494695, 1121212121213218605806, 122333333333334430727018, 13253646464646465743858331, 1427395060606060607157999745, 152942546575757575758673151260, 16314558708191919191920289312876

Offset: 1

Jonathan Vos Post, May 11 2006

			a(2) = 1 + 22 = 23 is prime.
a(6) = 1 + 22 + 333 + 4444 + 55555 + 666666 = 727021 is prime.
For what value of n is the next prime a(n)?
a(158), which has 474 digits, is prime. - _Harvey P. Dale_, Oct 17 2011

F. Smarandache, "Properties of the numbers", Univ. of Craiova Archives, 1975; Arizona State University Special Collections, Tempe, AZ.

Cf. A000461 (concatenate n n times), A004216, A048376, A053422.

Accumulate[FromDigits/@Table[Flatten[IntegerDigits/@PadLeft[{},n,n]], {n,15}]] (* Harvey P. Dale, Oct 17 2011 *)

a(n) = Sum_{i=1..n} A000461(i). a(n) = Sum_{i=1..n} i*(10^(i*L(i))-1)/(10^L(i)-1) where L(i) = A004216(i) + 1 = floor(log_10(10i)).

A118117 Concatenate n F(n) times.

Original entry on oeis.org

1, 2, 33, 444, 55555, 66666666, 7777777777777, 888888888888888888888, 9999999999999999999999999999999999, 10101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010101010

Offset: 1

Jonathan Vos Post, May 11 2006

A000461 Concatenate n n times.

			a(6) = 6 concatenated F(6) times = 6 concatenated 8 times = 66666666, where F(n) = the n-th Fibonacci number.

F. Smarandache, "Properties of the numbers", Univ. of Craiova Archives, 1975; Arizona State University Special Collections, Tempe, AZ.

Cf. A000045, A000461, A004216, A048376, A053422.

Table[FromDigits[Flatten[IntegerDigits/@PadRight[{},Fibonacci[n],n]]],{n,10}] (* Harvey P. Dale, Aug 09 2020 *)

a(n) = n concatenated A000045(n) times. a(n) = A000027(n) concatenated A000045(n) times.

cp's OEIS Frontend

A057692 Smallest prime which produces exactly n+1 different primes after n applications of the A048376 transform.

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A000461 Concatenate n n times.

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A053422 n times (n 1's): a(n) = n*(10^n - 1)/9.

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A048377 Append d digits d after each digit d in decimal expansion of n.

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A057628 Primes such that replacing each digit d with d copies of the digit d produces a prime. Zeros are not allowed.

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A057630 Primes such that replacing each digit d with d copies of the digit d produces a prime. Zeros are allowed.

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A158849 a(10n+m) is the integer with n+m concatenations of the digit m in base 10, 0<=m<=9.

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