A085264 Smallest number with exactly n representations as sum of a squarefree number (A005117) and a square (A000290).

1, 2, 6, 11, 23, 30, 38, 62, 71, 83, 110, 138, 155, 182, 203, 227, 263, 302, 327, 383, 435, 447, 503, 542, 602, 635, 707, 755, 798, 878, 915, 983, 1055, 1118, 1182, 1295, 1343, 1403, 1463, 1547, 1643, 1722, 1778, 1883, 1995, 2063, 2162, 2238, 2327

Offset

0,2

Comments

A085263(a(n))=n and A085263(i)<>n for i<a(n).

From Robert G. Wilson v, May 17 2014: (Start)

First occurrence of k, beginning with 0, in A085263.

Conjecture: Just as there is a least integer that can be represented in n ways, so is there a greatest integer.

Conjecture: The last occurrence of k, beginning with 0, in A085263: 13, 61, 85, 196, 225, 441, 621, 909, 1089, 1125, 1521, 2025, 2700, 2200, 2925, 3969, 3825, 4500, 5625, 4869, 6084, 8100, 11025, 7425, 9900, 9981, 10584, 11925, 12825, 14400, 13500, 14625, 18081, 18225, 17424, 20925, 22500, 27225, 21825, 25425, 27000, 28224, 27900, 38025, 44100, 33300, 35721, 35325, 39825, 37044, 39600, 40725, 44325, 55125, 50625, 53100, 52200, 54000, 60300, 65025, 63900, 60025, 63504, 64125, 74529, 81225, 77400, 99225, 88200, 76500, 79200, 87525, 90000, 108900, 88425, 91800, 95400, 96300, 100125, 107325, 132300, ..., .

Conjecture: For each j, there is a finite number of positive integers that can be represented as the sum of a squarefree number and a square in exactly j ways; e.g., for j=0, only the two integers 1 and 13 cannot be represented as the sum of a squarefree number and a square.

The number of integers that can be represented as the sum of a squarefree number and a square in j ways beginning with 0: 2, 9, 19, 27, 38, 36, 57, 63, 62, 74, 94, 86, 101, 112, 123, 113, 139, 140, 146, 170, 155, 202, 167, 196, 204, 213, 213, 215, 233, 232, 255, 249, 276, 261, 278, 310, 321, 300, 302, 336, 347, 325, 325, 350, 375, 367, 413, 393, 377, 384, 427, 435, 440, 447, 434, 472, 445, 476, 470, 518, 482, 499, 510, 542, 519, 550, 506, 553, 591, 572, 626, 586, 582, 585, 598, 623, 623, 656, 595, 697, 641, 672, 702, 689, 733, 696, 661, 718, 738, 757, 755, 739, 820, 734, 717, 834, 792, 811, 780, 831, 867, ..., .

(End)

Links

Table of n, a(n) for n=0..48.

Mathematica

f[n_] := f[n] = Count[ SquareFreeQ@# & /@ (n - Range[ Floor[ Sqrt[n]]]^2), True]; t = Array[ f, 10000]; Table[ Position[ t, n, 1, 1], {n, 0, 100}] (* Robert G. Wilson v, May 17 2014 *)

Crossrefs

Cf. A005117, A000290, A085263.

Sequence in context: A363936 A083322 A073939 * A005999 A102429 A080012

Adjacent sequences: A085261 A085262 A085263 * A085265 A085266 A085267

Keyword

nonn

Author

Reinhard Zumkeller, Jun 23 2003

Extensions

Edited by N. J. A. Sloane, May 23 2014

Status

approved