This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
Divisors[400] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2012 *)
divisors(400) \\ Charles R Greathouse IV, Jun 21 2017
Since the sum of the interior angles of a triangle is 180 degrees and an interior angle is 60 degrees on average, arithmetic sequences 60 - d, 60, 60 + d are possible, for integers d with 0 <= d <= 59. Therefore 3 is a term. Since the sum of the interior angles of a quadrilateral is 360 degrees and an interior angle is 90 degrees on average, arithmetic sequences 90 - 3d/2, 90 - d/2, 90 + d/2, 90 + 3d/2 are possible, for even d with 0 <= d <= 58. Therefore 4 is a term. Since the sum of the interior angles of a 16-gon is 2520 degrees and an interior angle is 157.5 degrees on average, arithmetic sequences 157.5 - 15d/2, 157.5 - 13d/2, 157.5 - 11d/2, 157.5 - 9d/2, 157.5 - 7d/2, 157.5 - 5d/2, 157.5 - 3d/2, 157.5 - d/2, 157.5 + d/2, 157.5 + 3d/2, 157.5 + 5d/2, 157.5 + 7d/2, 157.5 + 9d/2, 157.5 + 11d/2, 157.5 + 13d/2, 157.5 + 15d/2 are possible, for odd d with 1 <= d <= 19. Therefore 16 is a term.
Since A371663(17) = 45 and from a 45-gon 8 arithmetic progressions p_i(k) can formed from all its interior angles (all integer, in degrees), a(17) = 8. The 8 sequences are: p_1(k) = 172, p_2(k) = k + 150, p_3(k) = 2k + 128, p_4(k) = 3k + 106, p_5(k) = 4k + 84, p_6(k) = 5k + 62, p_7(k) = 6k + 40, p_8(k) = 7k + 18, for integers k with 0 <= k <= 44. Since A371663(19) = 60 and from a 60-gon 3 arithmetic progressions p_i(k) can formed from all its interior angles (all integer, in degrees), a(19) = 3. The 3 sequences are: p_1(k) = 174, p_2(k) = 2k + 115, p_3(k) = 4k + 56, for integers k with 0 <= k <= 15. Since A371663(10) = 16 and from a 16-gon 10 arithmetic progressions p_i(k) can formed from all its interior angles (all integer, in degrees), a(10) = 10. The 10 sequences are: p_1(k) = k + 150, p_2(k) = 3k + 135, p_3(k) = 5k + 120, p_4(k) = 7k + 105, p_5(k) = 9k + 90, p_6(k) = 11k + 75, p_7(k) = 13k + 60, p_8(k) = 15k + 45, p_9(k) = 17k + 30, p_10(k) = 19k + 15 for integers k with 0 <= k <= 15.
A371664:=proc(n) local a,L; L:=[3,4,5,6,8,9,10,12,15,16,18,20,24,30,36,40,45,48,60,72,80,90,120,144,180,240,360]; if (L[n]-2)*180/L[n]=floor((L[n]-2)*180/L[n]) then if L[n] mod 2 = 1 then a:=ceil(((L[n]-2)*360/L[n])/(L[n]-1)) else a:=ceil(((L[n]-2)*180/L[n])/(L[n]-1)) fi; elif (L[n]-2)*360/L[n]=floor((L[n]-2)*360/L[n]) and L[n] mod 2 = 0 then a:=ceil(((L[n]-2)*360/L[n]-L[n]+1)/(2*(L[n]-1))) fi; return a; end proc; seq(A371664(n),n=1..27);
a(3)=108, i.e., the value (in degrees) of the interior angles of the third polygon whose angles are integers (the pentagon).
a(3)=13 because the third polygon whose interior angle in degrees is not an integer is the 13-gon.
Select[Range[86],!Divisible[360,#] &] (* Stefano Spezia, Sep 05 2024 *)
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