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

Author

Jason Earls, Jul 03 2001

Keywords

• easy
• nonn
• base
• nice

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

Examples

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Index entries for primes involving decimal expansion of n

Crossrefs

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

Programs

Haskell

import Data.List (isInfixOf)
a062584 n = head [p | p <- a000040_list, show n `isInfixOf` show p]
-- Reinhard Zumkeller, Dec 29 2011

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 *)

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

Formula

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

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