A100952 Numbers that cannot be written as p*q+r with three distinct primes p, q and r.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 30, 36, 42, 60

Offset: 1

Author

Reinhard Zumkeller, Nov 23 2004

Keywords

• nonn

Comments

A100951(a(n)) = 0;

Conjecture: the sequence is complete.

A weaker conjecture: every integer greater than 60 (or some larger value based on further search) may be partitioned into a prime p and a semiprime qr, where the prime p is bounded by log(min(q,r)). Chen (1978) showed that all sufficiently large even numbers are the sum of a prime and the product of at most two primes. Zumkeller's conjecture effectively extends this from "even" to both even and odd integers. - Jonathan Vos Post, Nov 25 2004

Conjecture: Every positive integer can be represented as p*q-r with distinct primes p, q, r. - Zak Seidov, Aug 28 2012

Examples

			A100949(60) = #{11+7*7, 5+5*11, 3+3*19, 2+2*29} = 4, but A100951(60) = 0 as in each partition only 2 primes are used, therefore 60 is a term.
		

References

• Chen, J.-R. "On the Representation of a Large Even Number as the Sum of a Prime and the Product of at Most Two Primes, II." Sci. Sinica 21, 421-430, 1978.

Crossrefs

Cf. A100949, A100951.