A352030 -id:A352030 - cp's OEIS frontend

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

Hartmut F. W. Hoft, Mar 29 2022

nonn

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.

			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}
...

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

(* 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]

A375611 Numbers k whose symmetric representation of sigma(k) has at least a part with maximum width 2.

Original entry on oeis.org

6, 12, 15, 18, 20, 24, 28, 30, 35, 36, 40, 42, 45, 48, 54, 56, 63, 66, 70, 75, 77, 78, 80, 88, 91, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 130, 132, 135, 138, 143, 150, 153, 154, 156, 160, 162, 165, 170, 174, 175, 176, 182, 186, 187, 189, 190, 192, 195, 196, 200

Offset: 1

Hartmut F. W. Hoft, Aug 21 2024

nonn

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.

			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.
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.
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:
a(n)| row in A341970      left half of row in A341969
6   | 1   3               1   2
12  | 1   3               1   2
15  | 1   2   3   5       1   0   1   2
18  | 1   3   4           1   2   1
20  | 1   5               1   2
24  | 1   3               1   2
28  | 1   7               1   2
30  | 1   3   4   5       1   2   1   2
35  | 1   2   5   7       1   0   1   2
36  | 1   3   8           1   2   1
...

Column 2 of A253258.
Subsequence of A005279.
Some subsequences are A352030, A370205, A370206, A370209.
Cf. A003056, A174905, A235791, A237593, A249223, A249351, A250068, A341969, A341970.

eP[n_] := If[EvenQ[n], FactorInteger[n][[1, 2]], 0]+1
sDiv[n_] := Module[{d=Select[Divisors[n], OddQ]}, Select[Union[d, d*2^eP[n]], #<=row[n]&]]
mW2Q[n_] := Max[FoldWhileList[#1+If[OddQ[#2], 1, -1]&, sDiv[n], #1<=2&]]==2
a375611[m_, n_] := Select[Range[m, n], mW2Q]
a375611[1, 200]

A352061 Numbers n = 2^m * q, m > 0 and q > 1 odd, where the smallest odd divisor p > 1 is the m-th Mersenne prime 2^(m+1) - 1.

Original entry on oeis.org

6, 18, 28, 30, 42, 54, 66, 78, 90, 102, 114, 126, 138, 150, 162, 174, 186, 196, 198, 210, 222, 234, 246, 258, 270, 282, 294, 306, 308, 318, 330, 342, 354, 364, 366, 378, 390, 402, 414, 426, 438, 450, 462, 474, 476, 486, 496, 498, 510, 522, 532, 534, 546, 558, 570, 582, 594

Offset: 1

Hartmut F. W. Hoft, Mar 04 2022

nonn

All numbers in the sequence are pseudoperfect numbers since n = Sum_{i=0..m-1} (2^i * q) + Sum_{i=0..m} (2^i * q/p).
This sequence is a subsequence of A005835. It contains all even perfect numbers (A000396).
The first pseudoperfect number not in this sequence is A005835(2) = 12 = 2^2 * 3  since 3 is the first, not the second Mersenne prime.
The first pseudoperfect number in this sequence that is not in A352030 is 90 = 2*3*3*5 since its symmetric representation of sigma consists of one part with maximum width 3.
Since p = 2^(m+1) - 1 < 2^(m+1) the maximum width of the symmetric representation of sigma(a(n)) is at least 2, for all n.

			a(2) = 18 = 2 * 9 =  2^1 * (2^2 - 1) * 3  and a(9) = 90 = 2^1 * (2^2 - 1) * 15 since 3 is Mersenne prime A000668(1).
a(51) = 532 = 2^2 * (2^3 - 1) * 19  since 7 is Mersenne prime A000668(2).
a(757) = 8128 = 2^6 * (2^7 - 1) = 2^6 * (2^A000043(4) - 1) = 2^6 * A000668(4) = A000396(4) is a perfect number.

Cf. A000043, A000396, A000668, A005835, A237593, A352030.

evenoddPartsQ[n_] := Module[{dL=Select[Divisors[n], OddQ], fL=First[FactorInteger[n]], evenE}, evenE=If[First[fL]==2, Last[fL], 0]; n/2^evenE>1&&dL[[2]]==2^(evenE+1)-1]
a352061[n_] := Select[Range[n], evenoddPartsQ]
a352061[600]

cp's OEIS Frontend

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.

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A375611 Numbers k whose symmetric representation of sigma(k) has at least a part with maximum width 2.

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A352061 Numbers n = 2^m * q, m > 0 and q > 1 odd, where the smallest odd divisor p > 1 is the m-th Mersenne prime 2^(m+1) - 1.

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