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A358240 Consider all invertible residues mod n. For each residue, find the smallest product of three primes (A014612) which is in that residue class mod n. a(n) is the greatest of these.

%I A358240 #20 Jan 08 2024 23:56:43
%S A358240 8,27,28,45,66,175,45,105,76,171,102,325,165,261,124,273,230,385,188,
%T A358240 369,268,255,175,475,284,549,436,477,285,1309,332,385,430,927,318,
%U A358240 1127,442,639,610,657,595,1075,742,805,724,637,646,1705,642,741,670,1005,885,1435,801,1705,1105,873,1004,2821,938,873,844
%N A358240 Consider all invertible residues mod n. For each residue, find the smallest product of three primes (A014612) which is in that residue class mod n. a(n) is the greatest of these.
%H A358240 Charles R Greathouse IV, <a href="/A358240/b358240.txt">Table of n, a(n) for n = 1..10000</a>
%H A358240 Ramachandran Balasubramanian, Olivier Ramaré, and Priyamvad Srivastav, <a href="https://arxiv.org/abs/2208.04031">Product of three primes in large arithmetic progressions</a>, arXiv:2208.04031 [math.NT], 2022.
%F A358240 A result of Balasubramanian, Ramaré, & Srivastav proves that a(n) < n^e for each e > 9/2 and large enough n depending on e.
%e A358240 The least product of 3 primes = 1 mod 3 is 28, while the least = 2 mod 3 is 8, so a(2) = 28.
%o A358240 (PARI) firstTri(m)=my(mod=m.mod); forprime(p=2,, if(mod%p==0, next); forprime(q=2,p, if(mod%q==0, next); forprimestep(r=2,q,m/p/q, return(p*q*r))))
%o A358240 a(n)=my(r=8); for(k=1,n-1, if(gcd(k,n)>1, next); r=max(firstTri(Mod(k,n)),r)); r
%Y A358240 All terms are in A014612.
%Y A358240 Cf. A038026, A085420.
%K A358240 nonn
%O A358240 1,1
%A A358240 _Charles R Greathouse IV_, Jan 18 2023
%E A358240 Corrected by _Charles R Greathouse IV_, May 10 2023