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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.
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%I #25 Nov 03 2023 06:29:58

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

%N 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.

%C a(9) <= 76340177205657727, a(10) <= 225096507194749219819. - _David A. Corneth_, Jan 12 2023

%H Nilotpal Kanti Sinha, <a href="https://mathoverflow.net/questions/414186/are-there-highly-composite-prime-gaps">Are there highly composite prime gaps?</a> Question in mathoverflow, with an answer by Terry Tao, Jan 19 2022.

%e a(1) = 7: trivially, the 3 composites 8 = 2^3, 9 = 3^2, 10 = 2*5, have at least one distinct prime factor;

%e a(2) = 19: 20 = 2^2*5, 21 = 3*7, 22 = 2*11 all have 2 distinct prime factors;

%e a(3) = 643: 644 = 2^2*7*23, 645 = 3*5*43, 646 = 2*17*19, 647 is prime.

%o (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, mi=0; break); mi=min(mo,mi)); if (mi>=k, print1(pp,", "); k++)); pp=p)};

%o a359636(10^7)

%Y Cf. A001359, A075590, A185032.

%K nonn,hard,more

%O 1,1

%A _Hugo Pfoertner_, Jan 12 2023

%E a(8) from _Martin Ehrenstein_, Nov 03 2023