cp's OEIS Frontend

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.

A335574 Numbers of the form 16n^2 + 32n + 15 for which the central region of its symmetric representation of sigma consists of two subparts of sizes 4n+7 and 4n+1, n>=0.

Original entry on oeis.org

15, 63, 143, 255, 399, 575, 783, 1023, 1295, 1599, 1935, 2303, 2703, 3599, 4623, 5183, 6399, 7055, 7743, 8463, 9215, 9999, 10815, 11663, 12543, 16383, 17423, 18495, 19599, 20735, 21903, 23103, 24335, 25599, 26895, 28223, 29583, 32399, 36863, 38415, 39999
Offset: 0

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Author

Hartmut F. W. Hoft, Jan 26 2021

Keywords

Comments

The sequence is a subsequence of A141759. An alternate description is that for any divisor d <= row(a(k)) - see A235791 - of a(k) = (4n+3)(4n+5) the inequalities d != 4n+3, d != 4n+5 and 2d < 4n+3 hold in addition to 2*(4n+3) > row(a(k)). These conditions state that the symmetric representation of sigma consists of an odd number of regions and that the central region has maximum width 2. With the triangular function T in A235791 we get T[a(k), 4n+3] = T[(4n+3)(4n+5), 4n+3 ] = 2n + 4 and T[a(k), 4n+5] = T[(4n+3)(4n+5), 4n+5 ] = 2n + 1 determining the lengths of the two subparts - see A279387 - as 2*(2n+4) - 1 = 4n + 7 and 2*(2n+1) - 1 = 4n + 1 which results in the pattern [ 3 width 1, (4n + 1) width 2, 3 width 1 ] of unit cells and a total area of 8*(n+1) for the central region. The first region has area 8*(n+1)^2.

Examples

			a(3) = 255 = 3*5*17 = 15*17 = A141759(3) is in the sequence since 2*3 < 15 and 2*5 < 15 with row(255) = 22, and the central region of its symmetric representation of sigma has maximum width 2 and area 32 with subparts 4*3+7 = 19 and 4*3+1= 13.
3173 = 3*5*11*19 = 55*57 = A141759(13) is the first number in A141759 not in this sequence since the central region of the symmetric representation of sigma for 3173 has width 3 and also 2*(3*11) = 66 > 55.
a(37) = 32399 = 179*181 = A141759(44)  is in the sequence since the divisor conditions are vacuously true and the central region of its symmetric representation of sigma has maximum width 2 and area 8*45 = 360 with subparts 4*44 + 7 = 183 and 4*44 + 1 = 177.
35343 = 3*3*3*7*11*17 = (11*17)*(7*27) = 187*189 = A141759(46) is not in the sequence since 2*99, 2*119 and 2*153 exceed 187. While the area of the first region of its symmetric representation of sigma is 8*47^2 = 17672, the area of the central region is 21992 and of maximum width 5.
		

Crossrefs

Programs

  • Mathematica
    (* function segments[ ] is defined in A237270 *)
    centerQ[n_] := Module[{s=Select[segments[n], First[#]!=0&], len}, len=Length[s]; OddQ[len]&&Max[s[[(len+1)/2]]]==2]
    a335574[n_] := Select[Map[(4#+3)(4#+5)&, Range[0, n]], centerQ]
    a335574[50] (* sequence data *)
    (* alternative function based on divisors - much faster computation *)
    divisorQ[n_] := Module[{a=4n+3, b=4n+5, d, r}, r=Floor[(Sqrt[8 a b + 1] - 1)/2]; d=Select[Divisors[a b],#<=r&&#!=a&&#!=b&]; r<2a&&AllTrue[d, 2#
    				

Formula

a(k) = (4n+3)(4n+5) for n = sqrt(a(k)+1)/4 - 1, i.e., a(k) = A141759(n), for k>=0.