A359449 Positive integers in which the sum of the k-th powers of their digits is a prime number for k = 1, 2, 3, 4, 5, and 6 but not for k=7.
223, 232, 322, 1349, 1394, 1439, 1493, 1934, 1943, 2023, 2032, 2203, 2230, 2302, 2320, 3022, 3149, 3194, 3202, 3220, 3419, 3491, 3914, 3941, 4139, 4193, 4319, 4391, 4913, 4931, 9134, 9143, 9314, 9341, 9413, 9431, 10349, 10394, 10439, 10493, 10934, 10943, 13049, 13094, 13409, 13490, 13904, 13940
Offset: 1
Examples
223 belongs to this sequence because 2+2+3=7, 2^2+2^2+3^2=17, 2^3+2^3+3^3=43, 2^4+2^4+3^4=113, 2^5+2^5+3^5=307, and 2^6+2^6+3^6=857 are prime numbers whereas 2^7+2^7+3^7 is a composite number.
Programs
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Maple
filter:= proc(n) local L,t,k; L:= convert(n,base,10); andmap(isprime, [seq(add(t^k,t=L),k=1..6)]) and not isprime(add(t^7,t=L)) end proc: select(filter, [$1..20000]); # Robert Israel, Jan 03 2023
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Mathematica
For[a = 0, a <= 9, a++, For[b = 0, b <= 9, b++, For[c = 0, c <= 9, c++, For[d = 0, d <= 9, d++, If[PrimeQ[a + b + c + d] == True && PrimeQ[a^2 + b^2 + c^2 + d^2] == True && PrimeQ[a^3 + b^3 + c^3 + d^3] == True && PrimeQ[a^4 + b^4 + c^4 + d^4] == True && PrimeQ[a^5 + b^5 + c^5 + d^5] == True && PrimeQ[a^6 + b^6 + c^6 + d^6] == True && PrimeQ[a^7 + b^7 + c^7 + d^7] == False, Print[a, b, c, d]]]]]] (* This code outputs all the terms of the sequence in the interval [1,10^4]. *) -
PARI
isok(n) = my(d=digits(n)); for (i=1, 6, if (!isprime(sum(k=1,#d, d[k]^i)), return(0))); !isprime(sum(k=1,#d, d[k]^7)); \\ Michel Marcus, Jan 02 2023