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Numbers not of the form p + q^2 + r^3 + s^4 where p, q, r, and s are prime.
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%I #18 Nov 13 2018 09:06:52

%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,

%T 27,28,29,32,34,37,42,43,48,53,61,67,77,82,208

%N Numbers not of the form p + q^2 + r^3 + s^4 where p, q, r, and s are prime.

%C Probably finite.

%C There are no more terms < 10^11. - _Giovanni Resta_, Jun 10 2013

%H Liqun Hu and Li Yang, <a href="https://doi.org/10.1515/math-2017-0125">On pairs of equations in unlike powers of primes and powers of 2</a>, Open Mathematics 15:1 (2017), 8 pp.

%e 31 = 3 + 2^2 + 2^3 + 2^4 so 31 is not in the sequence. 32 cannot be written in a similar way so it is in the sequence.

%t max = 300; pqrs1234 = Sort[Flatten[Table[Prime[p] + Prime[q]^2 + Prime[r]^3 + Prime[s]^4, {p, PrimePi[max]}, {q, PrimePi[Sqrt[max]]}, {r, PrimePi[max^(1/3)]}, {s, PrimePi[max^(1/4)]}]]]; Complement[Range[max], pqrs1234] (* _Alonso del Arte_, Nov 24 2013 *)

%o (PARI) is(n)=if(n<30, return(n>0)); forprime(s=2,sqrtnint(n-14,4), my(lr=n-s^4); forprime(r=2,sqrtnint(lr-6,3), my(lq=lr-r^3); forprime(q=2,sqrtint(lq-2), if(isprime(lq-q^2), return(0))))); 1 \\ _Charles R Greathouse IV_, Nov 13 2018

%Y Cf. A226484.

%K nonn

%O 1,2

%A _Jud McCranie_, Jun 10 2013