login
A355660
Numbers m such that the smallest number of pentagonal numbers (A000326) which sum to m is exactly 4.
2
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
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
Equals A003679 \ A133929.
Sequence in context: A181310 A212110 A033310 * A312785 A312786 A312787
KEYWORD
nonn
AUTHOR
Bernard Schott, Jul 12 2022
STATUS
approved