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.

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

Author

Hartmut F. W. Hoft, Jan 26 2021

Keywords

• nonn

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

Cf. A141759, A235791, A237270, A237593, A279387.

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.