A176179 - OEIS

A176179

Primes such that the sum of digits, the sum of the squares of digits and the sum of 3rd powers of their digits is also a prime.

11, 101, 113, 131, 199, 223, 311, 337, 353, 373, 449, 461, 463, 641, 643, 661, 733, 829, 883, 919, 991, 1013, 1031, 1103, 1301, 1439, 1451, 1471, 1493, 1499, 1697, 1741, 1949, 2089, 2111, 2203, 2333, 2441, 2557, 3011, 3037, 3307, 3323, 3347, 3491, 3583, 3637, 3659, 3673, 3853, 4049, 4111, 4139, 4241, 4337, 4373, 4391, 4409

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OFFSET

1,1

COMMENTS

See A091365 for the exceptions for the case where the sum of the digits of p is not prime, but the sum of the cubes of the digits of p is prime.

REFERENCES

Charles W. Trigg, Journal of Recreational Mathematics, Vol. 20(2), 1988.

LINKS

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Mike Mudge, Morph code, Hands On Numbers Count, Personal Computer World, May 1997, p. 290.

EXAMPLE

For the prime number n =5693 we obtain :

5 + 6 + 9 + 3 = 23 ;

5^2 + 6^2 + 9^2 + 3^2 = 151 ;

5^3 + 6^3 + 9^3 + 3^3 = 1097.

MAPLE

with(numtheory):for n from 2 to 10000 do:l:=evalf(floor(ilog10(n))+1):n0:=n:s1:=0:s2:=0:s3:=0:for m from 1 to l do:q:=n0:u:=irem(q, 10):v:=iquo(q, 10):n0:=v :s1:=s1+u:s2:=s2+u^2:s3:=s3+u^3:od:if type(n, prime)=true and type(s1, prime)=true and type(s2, prime)=true and type(s3, prime)=true then print(n):else fi:od:

MATHEMATICA

okQ[n_]:=Module[{idn=IntegerDigits[n]}, And@@PrimeQ[Total/@{idn, idn^2, idn^3}]]; Select[Prime[Range[600]], okQ] (* Harvey P. Dale, Jan 18 2011 *)

PROG

(Python)

from sympy import isprime, primerange

def ok(p):

return all(isprime(sum(int(d)**k for d in str(p))) for k in [1, 2, 3])

def aupto(limit): return [p for p in primerange(1, limit+1) if ok(p)]

print(aupto(4409)) # Michael S. Branicky, Nov 23 2021

CROSSREFS

Cf. A109181, A046704, A052034, A091366.

Sequence in context: A073064 A155075 A180421 * A176196 A303570 A208262