cp's OEIS Frontend

A339272 a(n) is the smallest number k that can be partitioned into a set of n positive integers {e(1), e(2), ..., e(n)} where Sum_{i=1..n} e(i)*(e(i)-1) = k*(k-1)/2.

%I A339272 #20 Dec 15 2020 01:57:00
%S A339272 4,13,17,40,24,21,44,41,41,57,48,48,68,65,65,81,85,72,72,85,89,89,105,
%T A339272 109,109,96,116,133,113,113,133,133,140,140,120,157,153,137,157,164,
%U A339272 164,164,181,181,181,181,188,188,201,188,205,205,225,212,205,212,212,229,229,229,229,249,249,256,236,236,253,253,253
%N A339272 a(n) is the smallest number k that can be partitioned into a set of n positive integers {e(1), e(2), ..., e(n)} where Sum_{i=1..n} e(i)*(e(i)-1) = k*(k-1)/2.
%C A339272 These numbers solve the problem of what is the required minimum number of socks of n colors such that a random drawing of two socks has a 50% chance of matching. In this version there may be a color with only one sock, and the number of socks of each color need not be distinct.
%C A339272 If the set solving a(n)=k has three 2's, then a(n+1)<=k, replacing them with one 3 and three 1's; this leads to many repeats in the data. Similar substitutions include {4, 5} for {1, 2, 6}, {3, 3, 4} for {1, 2, 2, 5}, and {3, 3} for {1, 1, 4}. - _Dean D. Ballard_ and _Michael S. Branicky_, Nov 30 2020
%e A339272 For n = 4, {1, 2, 2, 12} is the set with the smallest sum that has this property. With 1 sock of one color, 2 socks of a second color, 2 socks of a third color, and 12 socks of a fourth color, there is exactly a 50% chance that a random draw of two socks will produce a matching pair. (1*0 + 2*1 + 2*1 + 12*11) = (17*16) / 2.
%e A339272 n = 2, sum = 4, set = {1, 3}
%e A339272 n = 3, sum = 13, set = {1, 3, 9}
%e A339272 n = 4, sum = 17, set = {1, 2, 2, 12}
%e A339272 n = 5, sum = 40, set = {3, 3, 3, 3, 28}
%e A339272 n = 6, sum = 24, set = {1, 1, 1, 2, 2, 17}
%o A339272 (PARI) \\ See 'Faster PARI Program' link in A246750 for PartsByWeight.
%o A339272 a(n)={local(FC=Map()); for(k=1, oo, if(PartsByWeight(n, k, k*(k-1)/2, (i,v)->v*(v-1)), return(k))); oo} \\ _Andrew Howroyd_, Nov 30 2020
%Y A339272 A246750, A332105, A339271 are other variations of the problem.
%K A339272 nonn
%O A339272 2,1
%A A339272 _Dean D. Ballard_, Nov 29 2020
%E A339272 a(26)-a(70) from _Andrew Howroyd_, Nov 30 2020