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%I #29 Jul 17 2021 11:22:47
%S 11,31,101,173,1307,1873,10111,11923,12209,14767,20357,20729,21149,
%T 22003,22151,29261,43681,43891,52033,52211,55231,58121,65011,70027,
%U 70399,80569,100087,101111,101401,102079,102113,120091,151931,163669,172001,200501,201113,203831
%N Numbers k such that replacing each digit d in the decimal expansion of k with d^3 yields a prime each time, when done recursively three times.
%H Harvey P. Dale, <a href="/A316982/b316982.txt">Table of n, a(n) for n = 1..235</a> (terms up to 2500000)
%e 173 is a term because replacing each digit d with d^3, recursively three times, a prime number is obtained: 173 -> 134327 (prime); 134327 -> 12764278343 (prime); 12764278343 -> 18343216648343512276427 (prime).
%e 1873 is a term because replacing each digit d with d^3, recursively three times, a prime number is obtained: 1873 -> 151234327 (prime); 151234327 -> 1125182764278343 (prime); 1125182764278343 -> 11812515128343216648343512276427 (prime).
%t A316982 = {}; Do[a=FromDigits[Flatten[IntegerDigits /@ (IntegerDigits[n]^3)]]; b=FromDigits[Flatten[IntegerDigits /@ (IntegerDigits[a]^3)]]; c=FromDigits[Flatten[IntegerDigits /@ (IntegerDigits[b]^3)]]; If[PrimeQ[a] && PrimeQ[b] && PrimeQ[c], AppendTo[A316982, n]], {n,300000}]; A316982 (* or *)
%t c[n_] := FromDigits@ Flatten@ IntegerDigits[IntegerDigits[n]^3]; Select[Range[204000], PrimeQ[x = c@#] && PrimeQ[y = c@x] && PrimeQ@c@y &] (* _Giovanni Resta_, Jul 18 2018 *)
%t p3[n_]:=Rest[NestList[FromDigits[Flatten[IntegerDigits/@(IntegerDigits[#]^3)]]&,n,3]]; Select[Range[205000],AllTrue[p3[#],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Aug 11 2019 *)
%o (PARI) eva(n) = subst(Pol(n), x, 10)
%o replace_digits(n) = my(d=digits(n), e=[]); for(x=1, #d, my(f=digits(d[x]^3)); if(f==[], e=concat(e, [0]), for(y=1, #f, e=concat(e, f[y])))); eva(e)
%o is(n) = my(x=n, i=0); while(i < 3, x=replace_digits(x); if(!ispseudoprime(x), break, i++)); i >= 3 \\ _Felix Fröhlich_, Oct 24 2018
%Y Cf. A048385, A048390, A048393, A316604.
%Y A004022 is a subsequence.
%K nonn,base
%O 1,1
%A _K. D. Bajpai_, Jul 18 2018
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