A085264 - OEIS

%I #13 Jul 19 2015 01:37:19

%S 1,2,6,11,23,30,38,62,71,83,110,138,155,182,203,227,263,302,327,383,

%T 435,447,503,542,602,635,707,755,798,878,915,983,1055,1118,1182,1295,

%U 1343,1403,1463,1547,1643,1722,1778,1883,1995,2063,2162,2238,2327

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

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

%C From _Robert G. Wilson v_, May 17 2014: (Start)

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

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

%C 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, ..., .

%C 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.

%C 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, ..., .

%C (End)

%t 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 *)

%Y Cf. A005117, A000290, A085263.

%K nonn

%O 0,2

%A _Reinhard Zumkeller_, Jun 23 2003

%E Edited by _N. J. A. Sloane_, May 23 2014