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.

8, 27, 28, 45, 66, 175, 45, 105, 76, 171, 102, 325, 165, 261, 124, 273, 230, 385, 188, 369, 268, 255, 175, 475, 284, 549, 436, 477, 285, 1309, 332, 385, 430, 927, 318, 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

Offset: 1

Author

Charles R Greathouse IV, Jan 18 2023

Keywords

• nonn

Examples

			The least product of 3 primes = 1 mod 3 is 28, while the least = 2 mod 3 is 8, so a(2) = 28.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Ramachandran Balasubramanian, Olivier Ramaré, and Priyamvad Srivastav, Product of three primes in large arithmetic progressions, arXiv:2208.04031 [math.NT], 2022.

Crossrefs

All terms are in A014612.

Cf. A038026, A085420.

Programs

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))))
a(n)=my(r=8); for(k=1,n-1, if(gcd(k,n)>1, next); r=max(firstTri(Mod(k,n)),r)); r

Formula

A result of Balasubramanian, Ramaré, & Srivastav proves that a(n) < n^e for each e > 9/2 and large enough n depending on e.

Extensions

Corrected by Charles R Greathouse IV, May 10 2023