A119748 - OEIS

%I #17 Mar 15 2023 05:25:48

%S 1,2,5,1549,1771561

%N Numbers that are not the sum of a prime and a (nontrivial, positive) power.

%C From a question raised by Tanya Khovanova.

%C 1771561 = 11^6 is the first composite number here. - _T. D. Noe_, Sep 29 2011

%C James Van Buskirk and John Robertson report that these are the only terms known up to 10^10. - _Charles R Greathouse IV_, Jan 30 2014

%C Hardy & Littlewood's Conjecture H implies that there are finitely many nonsquare terms in this sequence. Wang's result implies that a(n) >> n^1.018. - _Charles R Greathouse IV_, May 28 2015

%H G. H. Hardy, J. E. Littlewood, <a href="https://doi.org/10.1007/BF02403921">Some of the problems of partitio numerorum III: On the expression of a large number as a sum of primes</a>, Acta Mathematica 44 (1923), pp. 1-70.

%H John Robertson, <a href="http://mathforum.org/kb/message.jspa?messageID=1676583">Integers of the form x^2+kp</a> (1999). [Broken link]

%H Wang Tianze, <a href="https://doi.org/10.1007/BF02274058">On the exceptional set for the equation n = p + k^2</a>, Acta Mathematica Sinica, Vol. 11, No. 2 (1995), pp. 156-167.

%o (PARI) is(n)=for(e=2, log(n)\log(2), for(b=2, sqrtnint(n, e), if(isprime(n-b^e)&&!ispower(b), return(0)))); 1 \\ _Charles R Greathouse IV_, May 28 2015

%Y Cf. A119747.

%K nonn,more

%O 1,2

%A _David W. Wilson_, Jul 30 2006

%E 1771561 from _Max Alekseyev_, Jul 31 2006