A316982 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.

11, 31, 101, 173, 1307, 1873, 10111, 11923, 12209, 14767, 20357, 20729, 21149, 22003, 22151, 29261, 43681, 43891, 52033, 52211, 55231, 58121, 65011, 70027, 70399, 80569, 100087, 101111, 101401, 102079, 102113, 120091, 151931, 163669, 172001, 200501, 201113, 203831

Offset

1,1

Links

Harvey P. Dale, Table of n, a(n) for n = 1..235 (terms up to 2500000)

Example

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).
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).

Mathematica

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 *)
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 *)
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 *)

Prog

(PARI) eva(n) = subst(Pol(n), x, 10)
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)
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

Crossrefs

Cf. A048385, A048390, A048393, A316604.

A004022 is a subsequence.

Sequence in context: A299782 A027847 A068841 * A192246 A124296 A223388

Adjacent sequences: A316979 A316980 A316981 * A316983 A316984 A316985

Keyword

nonn,base

Author

K. D. Bajpai, Jul 18 2018

Status

approved