A275014 Upper ends of record gaps between numbers that are either primes or semiprimes.

3, 9, 29, 101, 247, 2531, 16033, 26615, 39359, 104677, 248426, 506531, 584237, 2560202, 4036993, 4417843, 5167619, 9725141, 25045807, 27489719, 70416301, 111555415, 174266734, 359589619, 1075714987, 6820213399, 15378035231, 16598109538, 19423306117, 30133946758, 74466436127

Offset: 1

Author

Bobby Jacobs, Nov 12 2016

Keywords

• nonn

Comments

This sequence is infinite, since the asymptotic density of the primes and semiprimes is 0. - Charles R Greathouse IV, Nov 12 2016

Examples

			a(5) = 247 because the next prime or semiprime after 241 is 247, and that is a record gap of size 6.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..37 (terms < 10^13)

Crossrefs

Cf. A037143, A111087, A275013, A275108.

Programs

PARI

r=0; last=2; for(n=3, 1e9, if(bigomega(n)<3, if(n-last>r, r=n-last; print1(n", ")); last=n)) \\ Charles R Greathouse IV, Nov 12 2016

PARI

checkrange(a, b, r)=while(b-a>r, forstep(n=a+r, a+1, -1, if(bigomega(n)<3, a=n; next(2))); for(n=a+r+1, b, if(bigomega(n)<3, return([a, n])))); 0
print1(3); p=5; r=1; forprime(q=7, 1e9, if(q-p<=r, p=q; next); t=checkrange(p, q, r); while(t!=0, print1(", "t[2]); t=checkrange(t[2], q, r=t[2]-t[1])); p=q) \\ Charles R Greathouse IV, Nov 12 2016

Formula

a(n) = A275013(n) + A275108(n).

Extensions

a(7)-a(31) from Charles R Greathouse IV, Nov 12 2016