A245475 Numbers n such that the sum of digits, sum of squares of digits, and sum of cubes of digits are all prime.

11, 101, 110, 111, 113, 131, 146, 164, 166, 199, 223, 232, 289, 298, 311, 322, 335, 337, 346, 353, 355, 364, 373, 388, 416, 436, 449, 461, 463, 494, 533, 535, 553, 566, 614, 616, 634, 641, 643, 656, 661, 665, 733, 829, 838, 883, 892, 919, 928, 944, 982, 991, 1001, 1010, 1011, 1013, 1031, 1046, 1064, 1066, 1099

Offset: 1

Author

Derek Orr, Jul 23 2014

Keywords

• nonn
• base

Comments

There are infinitely many numbers in this sequence; 0's can be added to any number any number of times in any logical order (i.e., the number doesn't start with a zero).

Examples

			1^1 + 4^1 + 6^1 = 11 is prime.
1^2 + 4^2 + 6^2 = 53 is prime.
1^3 + 4^3 + 6^3 = 281 is prime.
Thus 146, 164, 416, 461, 641, and 614 are members of this sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A182404, A225534, A108662.

Programs

Maple

filter:= proc(n) local L;
  L:= convert(n,base,10);
  isprime(convert(L,`+`)) and
  isprime(convert(map(`^`,L,2),`+`)) and
  isprime(convert(map(`^`,L,3),`+`))
end proc:
select(filter, [$1..2000]); # Robert Israel, Dec 04 2024

Mathematica

sdpQ[n_]:=Module[{idn=IntegerDigits[n]},AllTrue[{Total[idn], Total[ idn^2], Total[ idn^3]}, PrimeQ]]; Select[Range[1100],sdpQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 06 2018 *)

PARI

for(n=1,10^3,d=digits(n);s1=sum(i=1,#d,d[i]);s2=sum(j=1,#d,d[j]^2);s3=sum(k=1,#d,d[k]^3);if(isprime(s1)&&isprime(s2)&&isprime(s3),print1(n,", ")))