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Integers n such that appending some decimal digit to the first n digits of Pi results in a prime.
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%I #15 Aug 14 2022 13:24:17

%S 0,1,2,5,11,12,18,37,39,77,82,100,125,128,220,305,601,676,1692,1901,

%T 2202,2253,2394,3318,3970,5826,7001,9853,12607,13434,16207

%N Integers n such that appending some decimal digit to the first n digits of Pi results in a prime.

%C A140515 is a proper subsequence. A060421 - 1 is a proper subsequence. So the terms 47576 & 78072 are also members.

%e 0 is in the sequence since 2, 3, 5, and 7 are all primes;

%e 1 is in the sequence since 31 and 37 are both primes;

%e 2 is in the sequence since 311, 313, and 317 are all primes;

%e 3 is not in the sequence since 3141, 3143, 3147, and 3149 are all composites;

%e 4 is not in the sequence since 31411, 31413, 31417, and 31419 are all composites;

%e 5 is in the sequence since 314159 is a prime; etc.

%t fQ[n_] := Union[PrimeQ[ 10 IntegerPart[10^n*Pi] + {1, 3, 7, 9}]][[-1]]; k = -1; lst = {}; While[k < 17001, If[ fQ@ k, AppendTo[lst, k + 1]; Print[k + 1]]; k++]; lst

%t Module[{nn=16300,pd},pd=RealDigits[Pi,10,nn][[1]];Select[Range[0,nn],AnyTrue[ 10*FromDigits[Take[pd,#]]+{1,3,7,9},PrimeQ]&]] (* _Harvey P. Dale_, Aug 14 2022 *)

%o (PARI) is(n)=my(d=Pi*10^n\10*10);isprime(d+1) || isprime(d+3) || isprime(d+7) || isprime(d+9) \\ _Charles R Greathouse IV_, Nov 07 2013

%Y Cf. A140515, A060421, A005042.

%K nonn,base

%O 1,3

%A _Marvin Ray Burns_ and _Robert G. Wilson v_, Nov 07 2013

%E Keyword "base" added by _Zak Seidov_, Nov 11 2013