A355660 Numbers m such that the smallest number of pentagonal numbers (A000326) which sum to m is exactly 4.

4, 8, 16, 19, 20, 26, 30, 33, 38, 42, 50, 54, 60, 65, 67, 77, 81, 84, 88, 90, 96, 99, 100, 101, 111, 112, 113, 120, 125, 131, 135, 138, 142, 154, 159, 160, 166, 170, 171, 183, 195, 204, 205, 207, 217, 224, 225, 226, 229, 230, 236, 240, 241, 243, 255, 265, 275, 277, 286, 306, 308, 345

Offset: 1

Author

Bernard Schott, Jul 12 2022

Keywords

• nonn

Comments

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).

a(205) > 10^11, if it exists, from Giovanni Resta in A003679.

Examples

			4 = 1 + 1 + 1 + 1.
8 = 5 + 1 + 1 + 1.
16 = 5 + 5 + 5 + 1.
Also, it is not possible to get these terms when summing three or fewer pentagonal numbers.
		

References

• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section D3, Figurate numbers, pp. 222-228.

Links

• Richard K. Guy, Every number is expressible as the sum of how many polygonal numbers?, Amer. Math. Monthly 101 (1994), 169-172.

Crossrefs

Cf. A000326, A100878.

Equals A003679 \ A133929.

Programs

Mathematica

nn = 100;
pen = Table[n (3n - 1)/2, {n, 0, nn - 1}];
lst = Range[pen[[-1]]];
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}];
A003679 = lst;
Complement[A003679, {9, 21, 31, 43, 55, 89}] (* Jean-François Alcover, Jul 13 2022, after T. D. Noe in A003679 *)

Formula

A100878(a(n)) = 4.