A226484 -id:A226484 - cp's OEIS frontend

A212292 Odd numbers not of the form p^2 + q^2 + r with p, q, and r prime.

1, 3, 5, 7, 9, 17, 33, 43, 83, 179, 623, 713, 1019

Offset: 1

Charles R Greathouse IV, Jun 21 2012

The corresponding sequence with the restriction to primes removed is empty.
Wang shows that all but x^{9/20+e} members of this sequence up to x are congruent to 2 mod 3, for any e > 0.
There are no more terms < 10^7. - Donovan Johnson, Jun 27 2012
There are no more terms < 4*10^9. - Jud McCranie, Jun 09 2013
There are no more terms < 10^11. - Giovanni Resta, Jun 09 2013

Wang Mingqiang, On sums of a prime, and a square of prime, and a k-power of prime, Northeastern Mathematical Journal 18:4 (2002), pp. 283-286.

Cf. A045636, A006285, A118955, A156695, A226484.

list(lim)=my(p1=vector(primepi(sqrt(lim-5.5)),i,prime(i)^2), p2=List(), v=List(), u=List([1,3,5,7,9]), t); for(i=1,#p1, for(j=i,#p1,t=p1[i]+p1[j]; if(t>lim, break, listput(p2,t)))); p2=vecsort(Vec(p2),,8); for(i=1,#p2,forprime(p=2,lim-p2[i],listput(v,p2[i]+p))); v=select(n->n%2, vecsort(Vec(v),,8)); for(i=2,#v,forstep(j=v[i-1]+2,v[i]-2,2,listput(u,j))); Vec(u)

A226537 Numbers not of the form p + q^2 + r^3 + s^4 where p, q, r, and s are prime.

Original entry on oeis.org

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

Offset: 1

Jud McCranie, Jun 10 2013

nonn

Probably finite.

There are no more terms < 10^11. - Giovanni Resta, Jun 10 2013

			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.

Liqun Hu and Li Yang, On pairs of equations in unlike powers of primes and powers of 2, Open Mathematics 15:1 (2017), 8 pp.

Cf. A226484.

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 *)

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

cp's OEIS Frontend

A212292 Odd numbers not of the form p^2 + q^2 + r with p, q, and r prime.

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