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A352696 a(n) = k if the binary representation of k has a 1 (0) exactly where a 1 in the n-th row of A237048 occurs at an odd (even) position, reading from left to right.

%I A352696 #19 Mar 31 2023 14:22:29
%S A352696 1,1,2,1,2,3,2,1,5,2,2,3,2,2,11,1,2,6,2,3,10,2,2,3,5,2,10,3,2,13,2,1,
%T A352696 10,2,11,6,2,2,10,3,2,13,2,2,45,2,2,3,5,5,10,2,2,13,10,3,10,2,2,14,2,
%U A352696 2,43,1,10,13,2,2,10,11,2,7,2,2,44,2,11,12,2,3,21,2,2,14,10
%N A352696 a(n) = k if the binary representation of k has a 1 (0) exactly where a 1 in the n-th row of A237048 occurs at an odd (even) position, reading from left to right.
%C A352696 The phrase "symmetric representation of sigma(n)" is abbreviated below as SRS(n).
%C A352696 Every number in this sequence is a nondiving number and therefore in A061854. Number 22 with binary pattern 10110 is the smallest nondiving number in A061854, but not in this sequence since a number n with 5 odd divisors must have the form n = 2^m * p^4 for some prime p and some m>=0, and the pattern 10110 of odd/even positions of 1's in a row of A237048 requires 1's at positions 1 < 2^(m+1) < p < p^2 < 2^(m+1) * p <= row(n), a contradiction.
%C A352696 a(2^n) = 1 for all n>=0. The single part of SRS(2^n) has width 1, see A238443.
%C A352696 a(2^m * p) = 3 for odd primes p < 2^(m+1) with m >= 1. SRS(2^m * p) consists of a single part whose 2 subparts have sizes 2*T(n, 1) - 1 = 2^m * p - 1 and  2*T(n, p) - 1 = 2^m - p where T(n, k) = ceiling((n+1)/k -(k+1)/2), see A235791. The numbers 2^m * p are a subsequence of A174973 = A238443.
%C A352696 a(p^k) = A000975(k+1) for all odd primes p and k >= 0. Number a(p^k) in binary has k+1 digits with 1's and 0's alternating. SRS(p^k) has k+1 parts all of width 1 and of the symmetric sizes T(p^k, p^i) - T(p^k, 2*p^i) = (p^(k-i) + p^i)/2, for 0 <= i <= k. The numbers p^k are a subsequence of A174905, the odd primes p form the 1st column in the irregular triangle of A239929 and the numbers p^2 form the 1st column in the irregular triangle of A247687.
%e A352696 Sequence values for the first 4 powers of 3: {a(1), a(3), a(9), a(27)} = {1, 2, 5, 10} = {1, 10, 101, 1010}.
%e A352696 Table for a(1..16), a(27) and a(28) together with their lists of the base-2 representation, of the odd/even positions of 1's in the n-th row of A237048, and of the sizes of the parts in SRS(n):
%e A352696 n  a(n) odd/even   A237048         A237270
%e A352696 1   1   {1}        {1}             {1}
%e A352696 2   1   {1}        {1}             {3}
%e A352696 3   2   {1,0}      {1,1}           {2,2}
%e A352696 4   1   {1}        {1,0}           {7}
%e A352696 5   2   {1,0}      {1,1}           {3,3}
%e A352696 6   3   {1,1}      {1,0,1}         {12}
%e A352696 7   2   {1,0}      {1,1,0}         {4,4}
%e A352696 8   1   {1}        {1,0,0}         {15}
%e A352696 9   5   {1,0,1}    {1,1,1}         {5,3,5}
%e A352696 10  2   {1,0}      {1,0,0,1}       {9,9}
%e A352696 11  2   {1,0}      {1,1,0,0}       {6,6}
%e A352696 12  3   {1,1}      {1,0,1,0}       {28}
%e A352696 13  2   {1,0}      {1,1,0,0}       {7,7}
%e A352696 14  2   {1,0}      {1,0,0,1}       {12,12}
%e A352696 15 11   {1,0,1,1}  {1,1,1,0,1}     {8,8,8}
%e A352696 16  1   {1}        {1,0,0,0,0}     {31}
%e A352696 ...
%e A352696 27 10   {1,0,1,0}  {1,1,1,0,0,1}   {14,6,6,14}
%e A352696 28  3   {1,1}      {1,0,0,0,0,0,1} {56}
%e A352696 ...
%t A352696 (* function a237048[ ] is defined in A237048 *)
%t A352696 b237048[n_] := Fold[2#1+Mod[#2, 2]&, 0, Flatten[Position[a237048[n], 1]]]
%t A352696 a352696[n_] := Map[b237048, Range[n]]
%t A352696 a352696[85]
%Y A352696 Cf. A000975, A061854, A174905, A174973, A235791, A237048, A237270, A237591, A237593, A238443, A239929, A247687, A352030.
%K A352696 nonn
%O A352696 1,3
%A A352696 _Hartmut F. W. Hoft_, Mar 29 2022