A359636 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have at least n distinct prime factors.

7, 19, 643, 51427, 8083633, 1077940147, 75582271489, 34710483181813

Offset: 1

Author

Hugo Pfoertner, Jan 12 2023

Keywords

• nonn
• hard
• more

Comments

a(9) <= 76340177205657727, a(10) <= 225096507194749219819. - David A. Corneth, Jan 12 2023

Examples

			a(1) = 7: trivially, the 3 composites 8 = 2^3, 9 = 3^2, 10 = 2*5, have at least one distinct prime factor;
a(2) = 19: 20 = 2^2*5, 21 = 3*7, 22 = 2*11 all have 2 distinct prime factors;
a(3) = 643: 644 = 2^2*7*23, 645 = 3*5*43, 646 = 2*17*19, 647 is prime.
		

Links

• Nilotpal Kanti Sinha, Are there highly composite prime gaps? Question in mathoverflow, with an answer by Terry Tao, Jan 19 2022.

Crossrefs

Cf. A001359, A075590, A185032.

Programs

PARI

a359636(maxp) = {my (k=1, pp=3); forprime (p=5, maxp, my(mi=oo); if (p-pp>2, for (j=pp+1, p-1, my(mo=omega(j)); if (mo=k, print1(pp,", "); k++)); pp=p)};
a359636(10^7)

Extensions

a(8) from Martin Ehrenstein, Nov 03 2023