|
%I #30 Jul 14 2022 11:21:12
%S 4,8,16,19,20,26,30,33,38,42,50,54,60,65,67,77,81,84,88,90,96,99,100,
%T 101,111,112,113,120,125,131,135,138,142,154,159,160,166,170,171,183,
%U 195,204,205,207,217,224,225,226,229,230,236,240,241,243,255,265,275,277,286,306,308,345
%N Numbers m such that the smallest number of pentagonal numbers (A000326) which sum to m is exactly 4.
%C Richard Blecksmith & John Selfridge found 204 such integers among the first million, the largest of which is 33066. They believe that they have found them all (Richard K. Guy reference).
%C a(205) > 10^11, if it exists, from _Giovanni Resta_ in A003679.
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section D3, Figurate numbers, pp. 222-228.
%H Richard K. Guy, <a href="http://www.jstor.org/stable/2324367">Every number is expressible as the sum of how many polygonal numbers?</a>, Amer. Math. Monthly 101 (1994), 169-172.
%F A100878(a(n)) = 4.
%e 4 = 1 + 1 + 1 + 1.
%e 8 = 5 + 1 + 1 + 1.
%e 16 = 5 + 5 + 5 + 1.
%e Also, it is not possible to get these terms when summing three or fewer pentagonal numbers.
%t nn = 100;
%t pen = Table[n (3n - 1)/2, {n, 0, nn - 1}];
%t lst = Range[pen[[-1]]];
%t Do[n = pen[[i]]+pen[[j]]+pen[[k]]; If[n <= pen[[-1]], lst = DeleteCases[lst, n]], {i, 1, nn}, {j, i, nn}, {k, j, nn}];
%t A003679 = lst;
%t Complement[A003679, {9, 21, 31, 43, 55, 89}] (* _Jean-François Alcover_, Jul 13 2022, after _T. D. Noe_ in A003679 *)
%Y Cf. A000326, A100878.
%Y Equals A003679 \ A133929.
%K nonn
%O 1,1
%A _Bernard Schott_, Jul 12 2022
|
