A068679 - OEIS

%I #28 Aug 20 2024 01:56:57

%S 1,3,7,13,31,49,63,81,91,99,103,109,117,123,151,181,193,213,231,279,

%T 319,367,427,459,571,601,613,621,697,721,801,811,951,987,1113,1117,

%U 1131,1261,1821,1831,1939,2101,2149,2211,2517,2611,3151,3219,4011,4411,4519,4887,5031,5361,6231,6487,6871,7011,7209,8671,9141,9801,10051

%N Numbers which yield a prime whenever a 1 is inserted anywhere in them (including at the beginning or end).

%C If R(p) = (10^p-1)/9 is a prime then (10^(p-1)-1)/9 belongs to this sequence.

%H Giovanni Resta, <a href="/A068679/b068679.txt">Table of n, a(n) for n = 1..3314</a> (terms < 2*10^13, first 1929 terms from Chai Wah Wu)

%H C. Caldwell, <a href="http://www.utm.edu/research/primes/">Prime Pages</a>

%e 123 belongs to this sequence as the numbers 1123, 1213, 1231 obtained by inserting a 1 in all possible ways are all primes.

%t d[n_]:=IntegerDigits[n]; ins[n_]:=FromDigits/@Table[Insert[d[n],1,k],{k,Length[d[n]]+1}]; Select[Range[10060],And@@PrimeQ/@ins[#] &] (* _Jayanta Basu_, May 20 2013 *)

%t Select[Range[11000],AllTrue[FromDigits/@Table[Insert[ IntegerDigits[ #],1,n],{n,IntegerLength[#]+1}],PrimeQ]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Aug 16 2020 *)

%o (Python)

%o from sympy import isprime

%o A068679_list, n = [], 1

%o while len(A068679_list) < 1000:

%o     if isprime(10*n+1):

%o         s = str(n)

%o         for i in range(len(s)):

%o             if not isprime(int(s[:i]+'1'+s[i:])):

%o                 break

%o         else:

%o             A068679_list.append(n)

%o     n += 1 # _Chai Wah Wu_, Oct 02 2019

%Y Cf. A068673, A068674, A068677, A069246.

%K base,nonn

%O 1,2

%A _Amarnath Murthy_, Mar 02 2002

%E More terms from Eli McGowan (ejmcgowa(AT)mail.lakeheadu.ca), Apr 11 2002

%E More terms from _Vladeta Jovovic_, Apr 16 2002