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Numbers that are the sum of 3 nonzero squares in 4 or more ways.
+10
21
129, 134, 146, 153, 161, 171, 189, 194, 198, 201, 206, 209, 230, 234, 243, 246, 249, 251, 254, 257, 261, 266, 269, 270, 278, 281, 285, 290, 293, 294, 297, 299, 306, 314, 321, 326, 329, 339, 341, 342, 350, 353, 354, 362, 363, 365, 369, 371, 374, 378, 381, 386, 387, 389
CROSSREFS
Cf. A024796, A025295, A025322, A025323, A025324, A025325, A025326, A025327, A025328, A025329, A025330, A025331, A025332, A025333, A025334, A025335, A025336, A025337, A025338, A025369, A025427, A343968.
Numbers that are the sum of two positive cubes in at least four ways (all solutions).
+10
9
6963472309248, 12625136269928, 21131226514944, 26059452841000, 55707778473984, 74213505639000, 95773976104625, 101001090159424, 159380205560856, 169049812119552, 174396242861568, 188013752349696
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, D1.
LINKS
Uwe Hollerbach, Taxi, Taxi! [Replacement link to Wayback Machine] Uwe Hollerbach, Taxi! Taxi! [Cached copy from Wayback Machine, html version of top page only]
Numbers that are the sum of 2 nonzero squares in 5 or more ways.
+10
5
5525, 8125, 8450, 9425, 10625, 11050, 12025, 12325, 13325, 14365, 14450, 15725, 16250, 17225, 17425, 18125, 18785, 18850, 19825, 21125, 21250, 22100, 22525, 23125, 23725, 24050, 24505, 24650, 25625, 25925, 26650, 26825, 27625, 28730, 28925, 29725
MATHEMATICA
nn = 30000; t = Table[0, {nn}]; lim = Floor[Sqrt[nn - 1]]; Do[num = i^2 + j^2; If[num <= nn, t[[num]]++], {i, lim}, {j, i}]; Flatten[Position[t, _?(# >= 5 &)]] (* T. D. Noe, Apr 07 2011 *)
Numbers that are the sum of 2 distinct nonzero squares in 4 or more ways.
+10
5
1105, 1625, 1885, 2125, 2210, 2405, 2465, 2665, 3145, 3250, 3445, 3485, 3625, 3770, 3965, 4225, 4250, 4420, 4505, 4625, 4745, 4810, 4930, 5125, 5185, 5330, 5365, 5525, 5785, 5945, 6205, 6290, 6305, 6409, 6500, 6565, 6625, 6890, 6970, 7085, 7225, 7250
COMMENTS
Numbers in A025295 but not in A025314 are exactly those numbers of the form 2*p_1^(2*a_1)*p_2^(2*a_2)*...*p_m^(2*a_m)*q^6 where p_i are primes of the form 4k+3 and q is a prime of the form 4k+1. Thus 2*5^6 = 31250 is the smallest term in A025295 that is not in A025314. - Chai Wah Wu, Feb 27 2016
MATHEMATICA
nn = 7250; t = Table[0, {nn}]; lim = Floor[Sqrt[nn - 1]]; Do[num = i^2 + j^2; If[num <= nn, t[[num]]++], {i, lim}, {j, i - 1}]; Flatten[Position[t, _?(# >= 4 &)]] (* T. D. Noe, Apr 07 2011 *)
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