A332105
a(n) is the smallest number k that can be partitioned into a set of n distinct positive even integers {e(1), e(2), ..., e(n)} where the sum of e(i)*(e(i)-1) for i = 1 to n equals k*(k-1)/2.
Original entry on oeis.org
16, 144, 80, 96, 160, 208, 256, 304, 384, 432, 544, 608, 720, 816, 832, 1040, 1168, 1264, 1360, 1568, 1664, 1808, 1984, 2080, 2256, 2480, 2704, 2800, 3104, 3248, 3520, 3744, 3968, 4112, 4464, 4688, 4880, 5200, 5472, 5744, 6016, 6336, 6608, 6800, 7248, 7568, 7888, 8080, 8528
Offset: 2
For n = 3, {6, 48, 90} is the set of even numbers with the smallest sum that has this property. With 6 socks of one color, 48 socks of another color, and 90 socks of a third color, there is exactly a 50% chance that a random draw of two socks will produce a matching pair. (6*5 + 48*47 + 90*89) = (144*143) / 2.
n = 2, sum = 16, set = {6, 10}
n = 3, sum = 144, set = {6, 48, 90}
n = 4, sum = 80, set = {2, 8, 16, 54}
n = 5, sum = 96, set = {2, 6, 8, 14, 66}
n = 6, sum = 160, set = {2, 6, 8, 10, 24, 110}
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\\ See Links in A246750 for a faster program.
a(n)={for(k=n*(n+1)/2, oo, my(t=k*(4*k-1)); forpart(p=2*k-n*(n-1)/2, if(sum(i=1, n, (p[i]+i-1)*(2*(p[i]+i-1)-1))==t, return(4*k)), ,[n,n]))} \\ Andrew Howroyd, Nov 21 2020
A339271
a(n) is the smallest number k that can be partitioned into a set of n distinct positive integers {e(1), e(2), ..., e(n)} where Sum_{i=1..n} e(i)*(e(i)-1) = k*(k-1)/2.
Original entry on oeis.org
4, 13, 20, 53, 56, 92, 109, 120, 160, 200, 221, 268, 325, 389, 420, 497, 561, 616, 684, 725, 813, 901, 969, 1064, 1132, 1197, 1329, 1421, 1516, 1581, 1740, 1849, 1904, 2060, 2189, 2288, 2444, 2560, 2696, 2849, 2985, 3128, 3261, 3404, 3564, 3744, 3904, 4044, 4204, 4381, 4585, 4725
Offset: 2
For n = 3, {1, 3, 9} is the set with the smallest sum that has this property. With 1 socks of one color, 3 socks of another color, and 9 socks of a third color, there is exactly a 50% chance that a random draw of two socks will produce a matching pair. (1*0 + 3*2 + 9*8) = (13*12) / 2.
n = 2, sum = 4, set = {1, 3}
n = 3, sum = 13, set = {1, 3, 9}
n = 4, sum = 20, set = {1, 2, 3, 14}
n = 5, sum = 53, set = {1, 2, 3, 11, 36}
n = 6, sum = 56, set = {1, 2, 3, 5, 6, 39}
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\\ See 'Faster PARI Program' link in A246750 for PartsByWeight.
a(n)={local(FC=Map()); for(k=1, oo, if(PartsByWeight(n, k-n*(n-1)/2, k*(k-1)/2, (i,v)->(i+v-1)*(i+v-2)), return(k))); oo} \\ Andrew Howroyd, Nov 30 2020
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.
Original entry on oeis.org
4, 13, 17, 40, 24, 21, 44, 41, 41, 57, 48, 48, 68, 65, 65, 81, 85, 72, 72, 85, 89, 89, 105, 109, 109, 96, 116, 133, 113, 113, 133, 133, 140, 140, 120, 157, 153, 137, 157, 164, 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
Offset: 2
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.
n = 2, sum = 4, set = {1, 3}
n = 3, sum = 13, set = {1, 3, 9}
n = 4, sum = 17, set = {1, 2, 2, 12}
n = 5, sum = 40, set = {3, 3, 3, 3, 28}
n = 6, sum = 24, set = {1, 1, 1, 2, 2, 17}
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\\ See 'Faster PARI Program' link in A246750 for PartsByWeight.
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
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