Missing gnu's

[gnufinder]

Inspired by this site : https://www.math.auckland.ac.nz/~obrien/research/gnu.pdf I determined the numbers m upto 500, which are not among the gnu-values based on the currently known gnu's upto 50000. In other words, there is no n upto 50000 for which gnu(n)=m is known to be true. The first such number n is 33. The given link shows the moa's upto 100. If the value is not proven, a guess is made (the smallest cubefree number with the property). Only the guess for 73 is the smallest squarefree number with the property.

The "missing gnu's" are :

gap> Filtered([1..500],s->Size(Filtered([1..50000],t->gnu[t]=s))=0);
[ 33, 65, 82, 95, 107, 127, 133, 135, 146, 171, 190, 191, 206, 207, 212, 232,
  249, 263, 266, 273, 277, 280, 285, 286, 287, 291, 292, 294, 301, 305, 307,
  308, 317, 324, 325, 326, 327, 331, 337, 339, 341, 342, 344, 345, 346, 347,
  349, 350, 351, 352, 354, 356, 358, 359, 362, 363, 365, 366, 367, 368, 369,
  373, 375, 379, 381, 382, 383, 384, 385, 389, 392, 393, 394, 395, 396, 401,
  403, 406, 409, 410, 413, 414, 417, 419, 421, 422, 423, 425, 426, 428, 429,
  430, 435, 436, 437, 438, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451,
  453, 454, 459, 460, 461, 464, 471, 472, 473, 475, 476, 478, 479, 480, 481,
  482, 483, 484, 485, 486, 487, 490, 491, 494, 495, 497, 499 ]
gap> Size(last);
133
gap>

If gnu(n) has one of those values, n must satisfy lb(n)<500 (https://github.com/alex-konovalov/gnu/issues/74) The values n upto 50000 with lb(n)<500 are :

gap> Filtered([1..50000],s->(lb(s)<=500) and (gnu[s]=fail));
[ 9604, 11250, 11907, 12500, 14406, 14640, 16875, 16928, 17040, 17500, 17520,
  17820, 18711, 18750, 18960, 19208, 19920, 20250, 21296, 21360, 21609,
  22113, 23280, 24010, 24240, 24720, 25392, 25680, 26160, 26250, 26620,
  26912, 27120, 27500, 27540, 27783, 28125, 28812, 29282, 30375, 30480,
  30752, 31440, 32500, 32880, 33614, 33750, 35152, 35760, 36015, 36240,
  36360, 37260, 37680, 39120, 39204, 39304, 39375, 39546, 40080, 40368,
  41067, 41250, 41520, 42320, 42500, 42960, 43218, 43440, 43659, 43750,
  43808, 43923, 44550, 44896, 45840, 46128, 46305, 46320, 46980, 47142,
  47160, 47280, 47400, 47500, 47760, 48020, 48438, 48750, 48840, 48944,
  49005, 49320, 49560, 49800, 49952 ]
gap> Size(last);
96
gap>

I also search moa(2016) (definition see the link above). The smallest squarefree number with gnu(m)=2016 is m=1607970. I do not know whether a smaller cubefree number with this property exists. I am also looking for moa(2017). In this case, I do not even know a number m with gnu(m)=2017. Such a number should exists because in the article in the link above, the existence of the moa-function has been verified upto 10,000,000 (Unfortunately, I found no details how this was done).

2016 and 2017 do not occur among the known gnu-values upto 50000.

[gnufinder]

gnu(259083319343897905)=2017 shows at least the existence of moa(2017).

[olexandr-konovalov]

@gnufinder: thanks, this is interesting! I think these kinds of conjectures are good motivations to decide which numbers to try first. Clearly, we are now less interested in such suggestions <10K since not much left there, but when it comes to 10-50K or 50+K then they certainly help.

I haven't yet made up my mind whether it's better to post this as an issue or on wiki, so decided to leave it open there for a while...