A062584 a(n) is the smallest prime whose digits include the digits of n as a substring.

101, 11, 2, 3, 41, 5, 61, 7, 83, 19, 101, 11, 127, 13, 149, 151, 163, 17, 181, 19, 1201, 211, 223, 23, 241, 251, 263, 127, 281, 29, 307, 31, 1321, 233, 347, 353, 367, 37, 383, 139, 401, 41, 421, 43, 443, 457, 461, 47, 487, 149, 503, 151, 521, 53, 541, 557, 563

Offset

0,1

Comments

a(0) = 101 is the first term where a(n) has two more digits than n. a(665808) = 106658081 is the first term where a(n) has three more digits than n. For which n does a(n) first have four more digits than n? Is lim sup a(n)/n infinite? - Charles R Greathouse IV, Jun 23 2017

Decide how many elements you want to find. Then for every prime, check all substrings to see if they are the first to be in some n. If you've found all values a(n), then you want to stop. - David A. Corneth, Jun 24 2017

Links

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

Index entries for primes involving decimal expansion of n

Formula

a(n) = prime(A082058(n)). - Giovanni Resta, Apr 29 2017

Example

0 first occurs in 101. 14 first occurs as a substring in 149.

Mathematica

Do[k = 1; While[ StringPosition[ ToString[Prime[k]], ToString[n]] == {}, k++ ]; Print[ Prime[k]], {n, 0, 62} ]
(* Second program *)
Function[s, Table[FromDigits@ FirstCase[s, w_ /; SequenceCount[w, IntegerDigits@ n] > 0], {n, 0, 56}]]@ IntegerDigits@ Prime@ Range[10^4] (* Michael De Vlieger, Jun 24 2017 *)
With[{prs=Prime[Range[300]]}, Table[SelectFirst[prs, SequenceCount[ IntegerDigits[#], IntegerDigits[ n]]>0&], {n, 0, 60}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 06 2018 *)

Prog

(Haskell)
import Data.List (isInfixOf)
a062584 n = head [p | p <- a000040_list, show n `isInfixOf` show p]
-- Reinhard Zumkeller, Dec 29 2011
(PARI) build(root, prefixLen, suffixLen)=my(v=List(), t); for(i=10^(prefixLen-1)\1, 10^prefixLen-1, t=eval(Str(i, root))*10^suffixLen; for(n=t, t+10^suffixLen-1, listput(v, n))); Vec(v)
buildLen(n, d)=my(v=[]); for(i=0, d, v=concat(v, build(n, i, d-i))); Set(v)
a(n)=if(n==0, return(101)); my(d, v); while(1, v=buildLen(n, d); for(i=1, #v, if(isprime(v[i]), return(v[i]))); d++) \\ Charles R Greathouse IV, Jun 23 2017
(PARI) first(n) = my(res = vector(n), todo = n, g); forprime(p = 2, , d = digits(p); for(i=1, #d, if(d[i]!=0, g = d[i]; if(res[g]==0, res[g]=p; todo--); for(j=1, #d-i, g=10*g+d[i+j]; if(g>n, next(2)); if(res[g]==0, res[g]=p; todo--)))); if(todo==0, return(concat([101], res)))) \\ David A. Corneth, Jun 23 2017
(Python)
from sympy import nextprime
def a(n):
    p, s = 2, str(n)
    while s not in str(p): p = nextprime(p)
    return p
print([a(n) for n in range(57)]) # Michael S. Branicky, Dec 02 2021

Crossrefs

Cf. A082058, A018800, A060386. Similar to but different from A068164. E.g., a(133) = 4133, but A068164(133) = 1033.

Sequence in context: A266460 A341634 A060386 * A085054 A084045 A085419

Adjacent sequences: A062581 A062582 A062583 * A062585 A062586 A062587

Keyword

easy,nonn,base,nice

Author

Jason Earls, Jul 03 2001

Extensions

More terms from Lior Manor, Jul 08 2001

Corrected by Larry Reeves (larryr(AT)acm.org), Jul 10 2001

Further correction from Reinhard Zumkeller, Oct 14 2001

Name clarified by Jon E. Schoenfield, Dec 04 2021

Status

approved