A085257 Number of ways to write n as sum of a powerful and a squarefree number.

0, 1, 1, 1, 1, 2, 2, 1, 2, 3, 4, 2, 1, 4, 4, 2, 2, 4, 4, 2, 3, 4, 5, 2, 2, 4, 5, 3, 3, 7, 5, 4, 4, 5, 6, 2, 6, 9, 8, 4, 4, 9, 6, 4, 4, 8, 8, 4, 5, 7, 8, 3, 4, 6, 8, 5, 4, 6, 7, 5, 4, 9, 7, 4, 6, 10, 8, 5, 7, 9, 9, 3, 7, 10, 10, 2, 6, 12, 8, 5, 4, 10, 11, 5, 6, 11, 12, 5, 6, 8, 9, 6, 8, 11, 11, 6

Offset

1,6

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Powerful Number.

Eric Weisstein's World of Mathematics, Squarefree.

Formula

a(n) > 0 for n > 1 (proof?).

a(A085258(n)) = n and a(i) <> n for i < A085258(n).

Example

a(10) = 3: 10 = 1 + 9 = A005117(1) + A001694(4) = 2 + 8 = A005117(2) + A001694(3) = 6 + 4 = A005117(5) + A001694(2).

Mathematica

With[{m = 100}, pow = Select[Range[m], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]; sqf = Select[Range[m], SquareFreeQ]; BinCounts[Select[Plus @@@ Tuples[{pow, sqf}], # <= m &], {1, m, 1}]] (* Amiram Eldar, Feb 12 2023 *)

Crossrefs

Cf. A001694, A005117, A085258.

Sequence in context: A071444 A232441 A328701 * A352889 A230507 A352888

Adjacent sequences: A085254 A085255 A085256 * A085258 A085259 A085260

Keyword

nonn

Author

Reinhard Zumkeller, Jun 23 2003

Status

approved