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%I #22 Apr 08 2023 01:24:26
%S 1,3,4,14,18,23,62,95,423,5339,12352,108359,129805,5334194,82007322,
%T 90401717,199671691,434184265,655956850,44438886071
%N Numbers k > 0 that set a new record for the closeness of (4/3)*Pi*k^3 to an integer.
%C Integer radii such that the volume of the corresponding sphere is closer to an integer than for any smaller integer radius.
%H David Consiglio, Jr., <a href="/A297839/a297839_1.py.txt">Python Program</a>
%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a297/A297839.java">Java program</a> (github)
%e k | (4/3)*Pi*k^3 | Deviation from integer
%e ---------------------------------------------------------------------------
%e 1 | 4.188790204786390... | 0.188790204786390...
%e 3 | 113.097335529232556... | 0.097335529232556...
%e 4 | 268.082573106329023... | 0.082573106329023...
%e 14 | 11494.040321933856861... | 0.040321933856861...
%e 18 | 24429.024474314232222... | 0.024474314232222...
%e 23 | 50965.010421636019109... | 0.010421636019109...
%e 62 | 998305.991926330990581... | 0.008073669009418...
%e 95 | 3591364.001828731970435... | 0.001828731970435...
%e 423 | 317036825.999590816501793... | 0.000409183498206...
%e 5339 | 637482653747.999839504336479... | 0.000160495663520...
%e 12352 | 7894060641354.000003942767448... | 0.000003942767448...
%e 108359 | 5329464512150064.999997849950689... | 0.000000215004931...
%e 129805 | 9161421693208264.000000035388795... | 0.000000035388795...
%e 5334194 | 635762677398025211698.999999995151941... | 0.000000004848058...
%o (PARI) closeness(n) = my(v=(4/3)*Pi*n^3); if(round(v) > v, return(round(v)-v), return(v-round(v)))
%o my(r=1, k=1, c=0); while(1, c=closeness(k); if(c < r, print1(k, ", "); r=c); k++)
%Y Cf. A066645, A135973, A254714, A297840.
%K nonn,hard,more
%O 1,2
%A _Felix Fröhlich_, Jan 07 2018
%E a(15)-a(19) from _Jon E. Schoenfield_, Jan 07 2018
%E a(20) from _David Consiglio, Jr._, Mar 14 2023