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COMMENTS
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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)
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