A375611 - OEIS

A375611

Numbers k whose symmetric representation of sigma(k) has at least a part with maximum width 2.

%I #18 Sep 06 2024 20:19:44

%S 6,12,15,18,20,24,28,30,35,36,40,42,45,48,54,56,63,66,70,75,77,78,80,

%T 88,91,96,99,100,102,104,105,108,110,112,114,117,130,132,135,138,143,

%U 150,153,154,156,160,162,165,170,174,175,176,182,186,187,189,190,192,195,196,200

%N Numbers k whose symmetric representation of sigma(k) has at least a part with maximum width 2.

%C Number m = 2^k * q, k >= 0 and q odd, is in this sequence precisely when for any divisor s <= A003056(m) of q there is at most one divisor t of q satisfying s < t <= min(2^(k+1) * s, A003056(m)), and at least one such pair s < t of successive odd divisors exists. Equivalently, row m of the triangle in A249223 contains at least one 2, but no number larger than 2.

%e a(4) = 18 has width pattern 1 2 1 2 1 in its symmetric representation of sigma consisting of a single part, and row 18 in the triangle of A249223 is 1 1 2 1 1.

%e a(9) = 35 has width pattern 1 0 1 2 1 0 1 in its symmetric representation of sigma consisting of 3 parts, and row 35 in the triangle of A249223 is 1 0 0 0 1 1 2.

%e Irregular triangle of rows a(n) in triangle of A341970, i.e. of positions of 1's in triangle of A237048, and for the corresponding widths to the diagonal in triangle of A341969:

%e a(n)| row in A341970 left half of row in A341969

%e 6 | 1 3 1 2

%e 12 | 1 3 1 2

%e 15 | 1 2 3 5 1 0 1 2

%e 18 | 1 3 4 1 2 1

%e 20 | 1 5 1 2

%e 24 | 1 3 1 2

%e 28 | 1 7 1 2

%e 30 | 1 3 4 5 1 2 1 2

%e 35 | 1 2 5 7 1 0 1 2

%e 36 | 1 3 8 1 2 1

%e ...

%t eP[n_] := If[EvenQ[n], FactorInteger[n][[1, 2]], 0]+1

%t sDiv[n_] := Module[{d=Select[Divisors[n], OddQ]}, Select[Union[d, d*2^eP[n]], #<=row[n]&]]

%t mW2Q[n_] := Max[FoldWhileList[#1+If[OddQ[#2], 1, -1]&, sDiv[n], #1<=2&]]==2

%t a375611[m_, n_] := Select[Range[m, n], mW2Q]

%t a375611[1, 200]

%Y Column 2 of A253258.

%Y Subsequence of A005279.

%Y Some subsequences are A352030, A370205, A370206, A370209.

%Y Cf. A003056, A174905, A235791, A237593, A249223, A249351, A250068, A341969, A341970.

%K nonn

%O 1,1

%A _Hartmut F. W. Hoft_, Aug 21 2024