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A085264 Smallest number with exactly n representations as sum of a squarefree number (A005117) and a square (A000290). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

Comments 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}] (From Robert G. Wilson v, May 17 2014)

CROSSREFS

Cf. A005117, A000290, A085263.

Sequence in context: A135348 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

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Last modified September 30 08:01 EDT 2014. Contains 247417 sequences.