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%I #5 Jan 24 2020 21:00:24
%S 1,2,6,30,210,2310,3570,4830,11550,30030,43890,111930,131670,510510,
%T 690690,870870,1021020,2459730,9699690,13123110,17160990,40750710,
%U 146006070,223092870,340510170,358888530,688677990,1462190730,2445553110,2911018110,6469693230
%N Numbers k with a record number of divisors d < sqrt(k) such that d + k/d is prime.
%C The corresponding record values are 0, 1, 2, 4, 8, 12, 13, 14, 15, 21, 24, 25, 29, 40, 41, 46, 49, 51, 70, 77, 88, 89, 90, 117, 120, 147, 153, 154, 155, 161, 263, ...
%C Apparently all the primorial numbers (A002110) are terms. The record values of terms that are primorial numbers are terms of A103787.
%e 2 has one divisor below sqrt(2), 1, such that 1 + 2/1 = 3 is prime.
%e 6 has 2 divisors below sqrt(6), 1 and 2, such that 1 + 6/1 = 7 and 2 + 6/2 = 5 are primes.
%e 30 has 4 divisors below sqrt(30), 1, 2, 3, and 5 such that 1 + 30/1 = 31, 2 + 30/2 = 17, 3 + 30/3 = 13 and 5 + 30/5 = 11 are primes.
%t divCount[n_] := DivisorSum[n, Boole @ PrimeQ[# + n/#] &, #^2 < n &]; seq = {}; dm = -1; Do[d1 = divCount[n]; If[d1 > dm, dm = d1; AppendTo[seq, n]], {n, 1,10^6}]; seq
%Y Cf. A002110, A093890, A103787, A161510.
%K nonn
%O 1,2
%A _Amiram Eldar_, Jan 23 2020