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Least number that is both the sum and product of n distinct positive integers.
1

%I #14 Aug 10 2021 12:17:13

%S 1,3,6,24,120,720,5040,40320,362880,3628800,39916800,479001600,

%T 6227020800,87178291200,1307674368000,20922789888000,355687428096000,

%U 6402373705728000,121645100408832000,2432902008176640000,51090942171709440000,1124000727777607680000

%N Least number that is both the sum and product of n distinct positive integers.

%C Each a(n) = n! except that a(2) = 1+2 = 3. For n > 0, only each integer >= A000217(n) is the sum of n distinct positive integers. For the integers that are products of these types, see below.

%F a(n) = A000142(n) for n = 1 and n > 2; a(2) = 3.

%F a(n) = max(A000142(n), A000217(n)).

%F E.g.f.: x*(2 + x - x^2)/(2*(1 - x)). - _Stefano Spezia_, Oct 11 2020

%e a(1) = 1 because we define sums and products as sum(m) := prod(m) := m for all integers m in this case where these normally-binary operations only have one operand.

%e a(3) = 6 because 6 = 1+2+3 = 1*2*3 (with all the distinct positive integers the same in the sum and the product only for this term and a(1)).

%e a(5) = 120 because 120 = 1+2+3+4+110 (= ... = 22+23+24+25+26) = 1*2*3*4*5.

%t Array[If[# <= 2, (#^2 - #)/2 &[# + 1], #!] &, 22] (* _Michael De Vlieger_, Oct 15 2020 *)

%t With[{nn=30},Rest[CoefficientList[Series[x (2+x-x^2)/(2(1-x)),{x,0,nn}],x] Range[0,nn]!]] (* _Harvey P. Dale_, Aug 10 2021 *)

%o (PARI) a(n) = if(n<1, , if(n==2, 3, n!))

%Y Cf. A000142, A000217.

%Y Cf. Products of k distinct positive integers: A000027 (k=1), A020725 (k=2), A080257 (k=3), A122181 (k=4).

%K nonn,easy

%O 1,2

%A _Rick L. Shepherd_, Oct 10 2020