show/hide this revision's text 5 deleted 768 characters in body

8003 406 7995 405 595 19.7118227 2.19323119 4.81599071

16003 19.7407407 2.1966932 4.82334827 15995 775 1226 20.6490323 2.13304737 4.84226669

24003 1225 20.6387097 2.13209118 4.83998588 23995 1147 1854 20.9267655 2.07484654 4.79526455

32003 1496 1853 20.9197908 2.07422357 4.79375621 31995 1495 2505 21.3923797 2.06219738 4.82402134

40003 1832 21.4013378 2.06311065 4.826108 39995 1831 3169 21.8356987 2.06061119 4.86416246

48003 21.8432551 2.06136319 4.86589867 47995 2166 3835 22.1620499 2.05603592 4.88843418

56003 2510 3834 22.1583564 2.05572506 4.88766328 55995 2509 4491 22.3119522 2.04075959 4.88108964

64003 2861 22.3176564 2.041308 4.88237466 63995 2860 5140 22.3708494 2.0214589 4.85946442

72003 3178 22.3758741 2.02193578 4.86058798 71995 3177 5823 22.6567023 2.02573 4.89117691

80003 22.6613157 2.02616261 4.89220133 79995 3524 6477 22.7023269 2.01086714 4.87414365

88003 3849 6476 22.7000568 2.01068387 4.87368161 87995 3848 7152 22.8638607 2.00822199 4.88461397

96003 4180 22.8677235 2.00857731 4.88546218 95995 4179 7821 22.9672249 2.00200097 4.88472427

104003 22.9708064 2.00232771 4.88550694 103995 4499 8502 23.1169149 2.00108766 4.89640883

112003 4832 8501 23.1151367 2.00094706 4.89605147 111995 4831 9169 23.1794288 1.99370968 4.89110442

120003 23.1825709 1.99399218 4.89178523 119995 5142 9859 23.3378063 1.99549068 4.90728006

128003 5473 9858 23.3362505 1.99536902 4.90696952 127995 5472 10528 23.388087 1.98881521 4.90180839

136003 23.3908991 1.98906491 4.90241326 135995 5782 11219 23.5217918 1.98992654 4.91477942

144003 11218 23.5204082 1.98981939 4.91450488 143995 6107 11894 23.5799902 1.98525049 4.91280683

152003 11893 23.5786802 1.98514948 4.9125476 151995 6432 12569 23.6323072 1.98063941 4.91039132

159995 6754 13246 23.6889251 1.97689452 4.9095789

................

2255995 83886 198114 26.8935818 1.83836181 4.93234827

2263995 84140 198860 26.9074756 1.83886659 4.93414753

2271995 84425 199575 26.9114007 1.83869159 4.93412114

2279995 84715 200285 26.9137107 1.83840792 4.93380136

2287995 84997 201003 26.9185383 1.83829786 4.93394575

2295995 85307 201693 26.9144971 1.83758386 4.93246737

2303995 85594 202406 26.9177162 1.83736731 4.93232239

2311995 85887 203113 26.9190331 1.83702257 4.93183152

2319995 86158 203842 26.9272151 1.83714786 4.93260091

2327995 86438 204562 26.9325412 1.83707979 4.93284955

2335995 86698 205302 26.9440472 1.83743466 4.93423232

2343995 87001 205999 26.9421616 1.83687781 4.93316519

2351995 87295 206705 26.9430666 1.8365129 4.93261174 12568 23.6310634 1.98054391 4.91014581

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EDIT: Sunday 20 June: For $ 119995 \leq x \leq 1911995 $ the counting function is within half a percent either side of $$ \frac{ 0.214840253 \;x \; (\log \log x)^{0.942491757}}{\log x} $$ So Prof. Monsky's suspicion that there is a lower bound of the same nature as the upper bound (just different constant) becomes very interesting.

ORIGINAL:The counting function looks better with an extra factor of $ \log \log n $ in the numerator. That is, the cumulative count of numbers $ \equiv 3 \pmod 8 $ up to some $x$ that have an odd number of your representations resembles $$ \frac{ C \;x \; \log \log x}{\log x} $$

I do not know how to keep this in columns.

  n   odd  even    n / odd       / log n    * log log n 

  3     1     0    3           2.73071768   0.256818066

8003 406 595 19.7118227 2.19323119 4.81599071

16003 775 1226 20.6490323 2.13304737 4.84226669

24003 1147 1854 20.9267655 2.07484654 4.79526455

32003 1496 2505 21.3923797 2.06219738 4.82402134

40003 1832 3169 21.8356987 2.06061119 4.86416246

48003 2166 3835 22.1620499 2.05603592 4.88843418

56003 2510 4491 22.3119522 2.04075959 4.88108964

64003 2861 5140 22.3708494 2.0214589 4.85946442

72003 3178 5823 22.6567023 2.02573 4.89117691

80003 3524 6477 22.7023269 2.01086714 4.87414365

88003 3849 7152 22.8638607 2.00822199 4.88461397

96003 4180 7821 22.9672249 2.00200097 4.88472427

104003 4499 8502 23.1169149 2.00108766 4.89640883

112003 4832 9169 23.1794288 1.99370968 4.89110442

120003 5142 9859 23.3378063 1.99549068 4.90728006

128003 5473 10528 23.388087 1.98881521 4.90180839

136003 5782 11219 23.5217918 1.98992654 4.91477942

144003 6107 11894 23.5799902 1.98525049 4.91280683

152003 6432 12569 23.6323072 1.98063941 4.91039132

159995 6754 13246 23.6889251 1.97689452 4.9095789

................

2255995 83886 198114 26.8935818 1.83836181 4.93234827

2263995 84140 198860 26.9074756 1.83886659 4.93414753

2271995 84425 199575 26.9114007 1.83869159 4.93412114

2279995 84715 200285 26.9137107 1.83840792 4.93380136

2287995 84997 201003 26.9185383 1.83829786 4.93394575

2295995 85307 201693 26.9144971 1.83758386 4.93246737

2303995 85594 202406 26.9177162 1.83736731 4.93232239

2311995 85887 203113 26.9190331 1.83702257 4.93183152

2319995 86158 203842 26.9272151 1.83714786 4.93260091

2327995 86438 204562 26.9325412 1.83707979 4.93284955

2335995 86698 205302 26.9440472 1.83743466 4.93423232

2343995 87001 205999 26.9421616 1.83687781 4.93316519

2351995 87295 206705 26.9430666 1.8365129 4.93261174

show/hide this revision's text 3 deleted 1 characters in body

EDIT: Sunday 20 June: For $ 119995 \leq x \leq 1879995 1911995 $ the counting function is within half a percent either side of $$ \frac{ 0.214548031 0.214840253 \;x \; (\log \log x)^{0.943938004}}{\log x)^{0.942491757}}{\log x} $$ So Prof. Monsky's suspicion that there is a lower bound of the same nature as the upper bound (just different constant) becomes very interesting.

ORIGINAL:The counting function looks better with an extra factor of $ \log \log n $ in the numerator. That is, the cumulative count of numbers $ \equiv 3 \pmod 8 $ up to some $x$ that have an odd number of your representations resembles $$ \frac{ C \;x \; \log \log x}{\log x} $$

I do not know how to keep this in columns.

  n   odd  even    n / odd       / log n    * log log n 

  3     1     0    3           2.73071768   0.256818066

8003 406 595 19.7118227 2.19323119 4.81599071

16003 775 1226 20.6490323 2.13304737 4.84226669

24003 1147 1854 20.9267655 2.07484654 4.79526455

32003 1496 2505 21.3923797 2.06219738 4.82402134

40003 1832 3169 21.8356987 2.06061119 4.86416246

48003 2166 3835 22.1620499 2.05603592 4.88843418

56003 2510 4491 22.3119522 2.04075959 4.88108964

64003 2861 5140 22.3708494 2.0214589 4.85946442

72003 3178 5823 22.6567023 2.02573 4.89117691

80003 3524 6477 22.7023269 2.01086714 4.87414365

88003 3849 7152 22.8638607 2.00822199 4.88461397

96003 4180 7821 22.9672249 2.00200097 4.88472427

104003 4499 8502 23.1169149 2.00108766 4.89640883

112003 4832 9169 23.1794288 1.99370968 4.89110442

120003 5142 9859 23.3378063 1.99549068 4.90728006

128003 5473 10528 23.388087 1.98881521 4.90180839

136003 5782 11219 23.5217918 1.98992654 4.91477942

144003 6107 11894 23.5799902 1.98525049 4.91280683

152003 6432 12569 23.6323072 1.98063941 4.91039132

159995 6754 13246 23.6889251 1.97689452 4.9095789

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