This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A239635 #42 Dec 17 2016 12:10:32 %S A239635 1,12,24,72,240,360,1440,1440,1440,8640,10080,15120,34560,45360,55440, %T A239635 60480,60480,166320,181440,211680,332640,332640,332640,665280,665280, %U A239635 665280,831600,907200,1663200,2494800,2661120,2661120 %N A239635 Common Sigma, Uncommon Clique Numbers: a(n) is the minimal s for which there exists a set of n pairwise relatively prime integers with a sigma value of s. %C A239635 The meaning of the name just involves a little wordplay. %C A239635 Common Sigma: Refers to an underlying set with a common sigma. %C A239635 Uncommon Clique: This set or "clique" (see link below) of numbers is "uncommon" because they are pairwise relatively prime and thus have no common factors. %C A239635 Graph Theory Description: %C A239635 Suppose we built an incidence graph for each positive integer s whereby each number which had sigma = s represented a vertex in the graph and the only edges were drawn between relatively prime numbers. For each s, we want to find the maximal clique (aka complete subgraph) size. %C A239635 a(n) would be the minimum s where the maximal clique = n. %C A239635 (Note: If a clique of size n exists for s, there exists a clique of any size < n, so any gaps in the series could be filled by S (e.g., 1440: terms 7-9).) %C A239635 The first 5 terms and their solution sets (plus member factors showing they are all relatively prime to each other): %C A239635 1 : 1 %C A239635 12 : 6 = 2 * 3, 11 = 11 %C A239635 24 : 14 = 2 * 7, 15 = 3 * 5, 23 = 23 %C A239635 72 : 46 = 2 * 23, 51 = 3 * 17, 55 = 5 * 11, 71 = 71 %C A239635 240 : 135 = 3^3 * 5, 158 = 2 * 79, 203 = 7 * 29, 209 = 11 * 19, 239 = 239 %C A239635 An exhaustive search was performed to show that these are in fact the minimal terms. %C A239635 Comment with b-file submission: %C A239635 I found a few useful optimizations: %C A239635 1) For a sigma's possible solution set, only consider the numbers which have 3 or fewer distinct prime factors. %C A239635 2) Check if there's a number which is a prime (power). Set that aside. (Call the count c1.) %C A239635 3) Set aside any numbers with two prime factors (If there are duplicates, say two numbers which are multiples of two, pick the smallest exponent. The larger prime factor has less of chance of being found elsewhere in the solution set). Call this count c2. %C A239635 4) Then, discard any 3-prime factor numbers which are not relatively prime with the above numbers. %C A239635 5) Of those remaining numbers, determine the set of distinct primes found amongst their factors. Call this set size pc. %C A239635 6) If (pc/3) + c1 + c2 > maxCliqueSize found, we have a possibility of adding to this sequence and should try to find a "sub-clique" (among the remaining 3-prime numbers) which is maxCliqueSize-c1-c2. %H A239635 Fred Schneider, <a href="/A239635/b239635.txt">Table of n, a(n) for n = 1..65</a> %H A239635 Fred Schneider, <a href="/A239635/a239635.txt">Solution sets for terms 1..65</a> %H A239635 Wikipedia, <a href="http://en.wikipedia.org/wiki/Divisor_function">Sigma</a> %H A239635 Wikipedia, <a href="http://en.wikipedia.org/wiki/Clique_(graph_theory)">Clique</a> %Y A239635 Cf. sigma-related sequences: A000203, A007368. %K A239635 nonn,hard %O A239635 1,2 %A A239635 _Fred Schneider_, Mar 22 2014