74 Jean-Marie De Koninck
25 {204475052375, 204475052425, 204475052500, 204475052525, 204475052550,
204475052600, 204475052625, 204475052700, 204475052725, 204475052975,
204475053025, 204475053050, 204475053075, 204475053125, 204475053175,
204475053200, 204475053325, 204475053625, 204475053750, 204475053775,
204475053850, 204475053900, 204475054050, 204475054075, 204475054100,
204475054125, 204475054400, 204475054425, 204475054450, 204475054725,
204475054750, 204475054800, 204475054825, 204475055175, 204475055200}
26 {554805820452, 554805820478, 554805820504, 554805820556}
27 {1505578023621, 1505578023756, 1505578023783, 1505578024809, 1505578024836,
1505578024917, 1505578024971, 1505578024998, 1505578025025, 1505578025052,
1505578025106, 1505578025457, 1505578025484, 1505578025781, 1505578026105}
28 {4086199301996, 4086199302052, 4086199302080, 4086199302136, 4086199302248,
4086199302276, 4086199302332, 4086199302976, 4086199303004}
29 {11091501630949, 11091501630978, 11091501631181, 11091501631210,
11091501631239, 11091501631268, 11091501631384, 11091501631413,
11091501631442, 11091501631848, 11091501632022, 11091501632312,
11091501632544, 11091501632631, 11091501633037}
30 {30109570412400, 30109570412430, 30109570412970, 30109570413240,
30109570413270, 30109570413330, 30109570413360, 30109570419960,
30109570419990, 30109570420170, 30109570420260, 30109570420380,
30109570420440, 30109570420500, 30109570420530, 30109570420560,
30109570420590, 30109570420620, 30109570420710,30109570420770,
30109570420800, 30109570420950, 30109570420980, 30109570421010,
30109570421250, 30109570421280}
31 {81744303089590, 81744303090117, 81744303090706, 81744303091171,
81744303091202, 81744303091264, 81744303091295, 81744303091419,
81744303091636, 81744303091760, 81744303092287, 81744303092318,
81744303092504, 81744303092535, 81744303092566, 81744303092628,
81744303092690, 81744303092721, 81744303097960, 81744303098022,
81744303098053, 81744303098766}
32 {221945984401280, 221945984401312, 221945984401344, 221945984401440,
221945984402368, 221945984402400, 221945984403168, 221945984403840,
221945984403872, 221945984403904, 221945984404000, 221945984404064,
221945984410016, 221945984410048, 221945984410272, 221945984410336,
221945984411104, 221945984411136, 221945984411200, 221945984411232,
221945984411264, 221945984411680, 221945984411712, 221945984411744}
observe that the set A11 contains only one element, while #Am 3 for
all 2 m 32, m = 11; it would be interesting if one could prove that
limm→∞ #Am = +∞ (or prove the contrary !); on the other hand, one can
prove79
that when k is large, then each nk Ak satisfies
nk = exp{(1 + o(1))(k + 1)}.
333
the largest number which cannot be written as the sum of six non zero distinct
squares (F.Halter-Koch): see R.K. Guy [101], C20;
the fourth term of the sequence (nk)k≥1 defined as follows: n1 = 1 and for
each k 2, nk is the smallest number n such that | sin n| | sin nk−1|, so
79Indeed,
this follows essentially from the fact that, according to the Prime Number Theorem,
π(x)
x
log x
+
x
log2
x
as x ∞.
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