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

1,1

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.

References

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

Links

Table of n, a(n) for n=1..62.

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

Formula

A100878(a(n)) = 4.

Example

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.

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 *)

Crossrefs

Cf. A000326, A100878.

Equals A003679 \ A133929.

Sequence in context: A181310 A212110 A033310 * A312785 A312786 A312787

Adjacent sequences: A355657 A355658 A355659 * A355661 A355662 A355663

Keyword

nonn

Author

Bernard Schott, Jul 12 2022

Status

approved