A119748 Numbers that are not the sum of a prime and a (nontrivial, positive) power.

1, 2, 5, 1549, 1771561

Offset: 1

Author

David W. Wilson, Jul 30 2006

Keywords

• nonn
• more

Comments

From a question raised by Tanya Khovanova.

1771561 = 11^6 is the first composite number here. - T. D. Noe, Sep 29 2011

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

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

Links

• G. H. Hardy, J. E. Littlewood, Some of the problems of partitio numerorum III: On the expression of a large number as a sum of primes, Acta Mathematica 44 (1923), pp. 1-70.
• John Robertson, Integers of the form x^2+kp (1999). [Broken link]
• Wang Tianze, On the exceptional set for the equation n = p + k^2, Acta Mathematica Sinica, Vol. 11, No. 2 (1995), pp. 156-167.

Crossrefs

Cf. A119747.

Programs

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

Extensions

1771561 from Max Alekseyev, Jul 31 2006