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.

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, 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, 2, 43, 1, 10, 13, 2, 2, 10, 11, 2, 7, 2, 2, 44, 2, 11, 12, 2, 3, 21, 2, 2, 14, 10

Offset: 1

Author

Hartmut F. W. Hoft, Mar 29 2022

Keywords

• nonn

Comments

The phrase "symmetric representation of sigma(n)" is abbreviated below as SRS(n).

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.

a(2^n) = 1 for all n>=0. The single part of SRS(2^n) has width 1, see A238443.

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.

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.

Examples

			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}.
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):
n  a(n) odd/even   A237048         A237270
1   1   {1}        {1}             {1}
2   1   {1}        {1}             {3}
3   2   {1,0}      {1,1}           {2,2}
4   1   {1}        {1,0}           {7}
5   2   {1,0}      {1,1}           {3,3}
6   3   {1,1}      {1,0,1}         {12}
7   2   {1,0}      {1,1,0}         {4,4}
8   1   {1}        {1,0,0}         {15}
9   5   {1,0,1}    {1,1,1}         {5,3,5}
10  2   {1,0}      {1,0,0,1}       {9,9}
11  2   {1,0}      {1,1,0,0}       {6,6}
12  3   {1,1}      {1,0,1,0}       {28}
13  2   {1,0}      {1,1,0,0}       {7,7}
14  2   {1,0}      {1,0,0,1}       {12,12}
15 11   {1,0,1,1}  {1,1,1,0,1}     {8,8,8}
16  1   {1}        {1,0,0,0,0}     {31}
...
27 10   {1,0,1,0}  {1,1,1,0,0,1}   {14,6,6,14}
28  3   {1,1}      {1,0,0,0,0,0,1} {56}
...
		

Crossrefs

Cf. A000975, A061854, A174905, A174973, A235791, A237048, A237270, A237591, A237593, A238443, A239929, A247687, A352030.

Programs

Mathematica

(* function a237048[ ] is defined in A237048 *)
b237048[n_] := Fold[2#1+Mod[#2, 2]&, 0, Flatten[Position[a237048[n], 1]]]
a352696[n_] := Map[b237048, Range[n]]
a352696[85]