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

1, 2, 5, 1549, 1771561

Offset

1,2

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

Table of n, a(n) for n=1..5.

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.

Prog

(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

Crossrefs

Cf. A119747.

Sequence in context: A078748 A051131 A306785 * A101341 A079014 A215436

Adjacent sequences: A119745 A119746 A119747 * A119749 A119750 A119751

Keyword

nonn,more

Author

David W. Wilson, Jul 30 2006

Extensions

1771561 from Max Alekseyev, Jul 31 2006

Status

approved