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

Author

Reinhard Zumkeller, Jun 23 2003

Keywords

• nonn

Comments

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

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)

Crossrefs

Cf. A005117, A000290, A085263.

Programs

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 *)

Extensions

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

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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