A057628 -id:A057628 - cp's OEIS frontend

A048376 Replace each 1 in decimal expansion of n with 1 1's, each 2 with 2 2's, etc. (0 vanishes).

1, 22, 333, 4444, 55555, 666666, 7777777, 88888888, 999999999, 1, 11, 122, 1333, 14444, 155555, 1666666, 17777777, 188888888, 1999999999, 22, 221, 2222, 22333, 224444, 2255555, 22666666, 227777777, 2288888888, 22999999999, 333, 3331

Offset: 1

Patrick De Geest, Mar 15 1999

			12 -> 122, 123->122333.

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

Cf. A048377, A057628, A057630.

With[{rules=Table[n->Table[n,{n}],{n,0,9}]},Table[FromDigits[ Flatten[ IntegerDigits[x]/.rules]],{x,40}]] (* Harvey P. Dale, Oct 09 2011 *)

A048376(n)={sum(i=1,#n=concat( apply( t->vector(t,i,t),digits(n) )),n[i]*10^(#n-i))} \\ 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

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

Original entry on oeis.org

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

A244853 Let d(1)d(2)... d(q) denote the decimal expansion of a prime number n > 9. The sequence lists the primes such that replacing each digit d(i) with d(i+1) copies for i = 1..q-1 and d(q) with d(1) copies produces a prime. Zeros are not allowed.

Original entry on oeis.org

11, 17, 71, 113, 131, 151, 167, 181, 211, 227, 281, 431, 467, 521, 547, 617, 743, 829, 853, 883, 1163, 1193, 1733, 2131, 2137, 3121, 3181, 3413, 3457, 3727, 4441, 5351, 6143, 6151, 6473, 6779, 6823, 6977, 8263, 8293, 8423, 9787, 11273, 11321, 11369, 11483

Offset: 1

Michel Lagneau, Jul 07 2014

			6473 is in the sequence because 6473 becomes 66664444444777333333 which is also prime.

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

Cf. A057628.

for n from 5 to 2500 do:
     p := ithprime(n): s :=0: j :=0:
     x := convert(p, base, 10): n1 := nops(x):
     q := mul(x[i], i=1..n1):
     if q<>0 then
         for m from n1 by -1 to 2 do:
            s := s*10^x[m-1]+add(x[m]*10^(i-1+j),i=1.. x[m-1]):
         od:
         s1 := add(x[1]*10^(i-1), i=1..x[n1]):
         z := s*10^x[n1]+s1:
         if isprime(z) then printf(`%d, `, p) fi:
     fi:
od:

deQ[n_]:=Module[{idn=IntegerDigits[n]},idn=Join[idn,{idn[[1]]}];FreeQ[ idn,0] && PrimeQ[FromDigits[Flatten[Table[#[[1]],{#[[2]]}]&/@ Partition[ idn,2,1]]]]]; Select[ Prime[Range[5,1500]],deQ] (* Harvey P. Dale, Mar 26 2016 *)

isok(n) = {if (isprime(n) && (d=digits(n)) && (#d>1) && vecmin(d), s = ""; for (id = 1, #d, if (id != #d, idk = d[id+1], idk = d[1]); for (k=1, idk, s = concat(s, d[id]));); isprime(eval(s)););} \\ Michel Marcus, Jul 09 2014

cp's OEIS Frontend

A048376 Replace each 1 in decimal expansion of n with 1 1's, each 2 with 2 2's, etc. (0 vanishes).

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

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PARI

Extensions

A244853 Let d(1)d(2)... d(q) denote the decimal expansion of a prime number n > 9. The sequence lists the primes such that replacing each digit d(i) with d(i+1) copies for i = 1..q-1 and d(q) with d(1) copies produces a prime. Zeros are not allowed.

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