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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. 1

%I

%S 15,63,143,255,399,575,783,1023,1295,1599,1935,2303,2703,3599,4623,

%T 5183,6399,7055,7743,8463,9215,9999,10815,11663,12543,16383,17423,

%U 18495,19599,20735,21903,23103,24335,25599,26895,28223,29583,32399,36863,38415,39999

%N 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.

%C 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.

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

%e 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.

%e 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.

%e 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.

%e 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.

%t (* function segments[ ] is defined in A237270 *)

%t centerQ[n_] := Module[{s=Select[segments[n], First[#]!=0&], len}, len=Length[s]; OddQ[len]&&Max[s[[(len+1)/2]]]==2]

%t a335574[n_] := Select[Map[(4#+3)(4#+5)&, Range[0, n]], centerQ]

%t a335574[50] (* sequence data *)

%t (* alternative function based on divisors - much faster computation *)

%t 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#<a&]]

%t a335574D[n_] := Map[(4#+3)(4#+5)&, Select[Range[0, n], divisorQ]]

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

%K nonn

%O 0,1

%A _Hartmut F. W. Hoft_, Jan 26 2021

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Last modified September 20 06:59 EDT 2021. Contains 347577 sequences. (Running on oeis4.)