A176196 Primes such that the sum of k-th powers of digits, for each of k = 1, 2, 3, and 4, is also a prime.

11, 101, 113, 131, 223, 311, 353, 461, 641, 661, 883, 1013, 1031, 1103, 1301, 1439, 1451, 1471, 1493, 1697, 1741, 2111, 2203, 3011, 3347, 3491, 3659, 4139, 4337, 4373, 4391, 4733, 4931, 5303, 5639, 5693, 6197, 6359, 6719, 6791, 6917, 6971, 7411, 7433

Offset: 1

Author

Michel Lagneau, Apr 11 2010

Keywords

• nonn
• base

Comments

For k = 1, 2, and 3 see A176179

Examples

			For the prime number n=14549 we obtain :
1 + 4 + 5 + 4 + 9 = 23 ;
1^2 +4^2 + 5^2 +4^2 + 9^2 = 139 ;
1^3 +4^3 + 5^3 +4^3 + 9^3 = 983 ;
1^4 +4^4 + 5^4 +4^4 + 9^4 = 7699 ;
		

References

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

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Mike Mudge, Morph code, Hands On Numbers Count, Personal Computer World, May 1997, p. 290.

Crossrefs

Cf. A176179, A109181, A046704, A052034, A091366.

Programs

Maple

with(numtheory):for n from 2 to 20000 do:l:=evalf(floor(ilog10(n))+1):n0:=n:s1:=0:s2:=0:s3:=0:s4:=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:s4:=s4+u^4:od:if type(n,prime)=true and type(s1,prime)=true and type(s2,prime)=true and type(s3,prime)=true and type(s4,prime)=true then print(n):else fi:od:

Mathematica

Select[Prime[Range[1000]],And@@PrimeQ[Total/@Table[IntegerDigits[#]^n,{n,4}]]&] (* Harvey P. Dale, Jun 16 2013 *)

Python

from sympy import isprime, primerange
def ok(p):
    return all(isprime(sum(int(d)**k for d in str(p))) for k in range(1, 5))
def aupto(limit): return [p for p in primerange(1, limit+1) if ok(p)]
print(aupto(7443)) # Michael S. Branicky, Nov 23 2021