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a(k) = count is the number of different widths patterns in the symmetric representation of sigma for numbers having k odd divisors.

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# div |count| first occurrence of distinct width patterns

# div |count| first occurrence of distinct width patterns

In the irregular triangle below, row k lists the count and the first occurrences of successive instantiations of the distinct width patterns in the symmetric representation of sigma for numbers with k odd divisors.

# div |count| first occurrence of distinct width patterns

| | 1 2 3 4 5 6 7 .. 11 .. 16 .. 40

-----------------------------------------------------------------------

1 | 1 | 1 . . .

2 | 2 | 3 6 . . .

3 | 3 | 9 18 72 . . .

4 | 6 | 15 21 30 60 78 120 . . .

5 | 5 | 81 162 648 1296 5184 . . .

6 | 16 | 45 63 75 90 147 150 180 ... 27744 .

7 | 7 | 729 1458 5832 11664 46656 93312 373248 . .

8 | 40 | 105 135 165 189 210 231 357 ... 203808

9 | 28? | 225 441 450 882 900 1225 1800 ...

10 | >=47| 405 567 810 1134 1377 1539 1620 ...

11 | 11 |59049 ... 1934917632

The complete sequence of first occurrences of the 11 width patterns for numbers with 11 odd divisors is: 59049, 118098, 472392, 944784, 3779136, 7558272, 30233088, 120932352, 241864704, 967458816, 1934917632.

The column labeled '1' of least occurrences of a width pattern of length 2k-1 is sequence A038547: least number with exactly k odd divisors.

t249223[n_] := FoldList[#1+(-1)^(#2+1)KroneckerDelta[Mod[n-#2 (#2+1)/2, #2]]&, 1, Range[2, Floor[(Sqrt[8n+1]-1)/2]]]

(* row n in triangle of A249223 *)

t262045[n_] := Join[t249223[n], Reverse[t249223[n]]] (* row n in triangle of A262045 *)

widthPattern[n_] := Map[First, Split[t262045[n]]]

nOddDivs[n_] := Length[Divisors[NestWhile[#/2&, n, EvenQ[#]&]]]

count[n_, k_] := Length[Union[Map[widthPattern, Select[Range[n], nOddDivs[#]==k&]]]]

(* count of distinct width patterns for numbers with k odd divisors in the range 1 .. n *)

Cf. A235791, A237048, A237270, A237591, A237593, A249223, A250071, A262045, A318843, A341969, A342592, A342594, A342595, A342596, A367377.

nonn,changed,more

allocated a(k) = count of different widths patterns in the symmetric representation of sigma for Hartmut F. Wnumbers having k odd divisors. Hoft

1, 2, 3, 6, 5, 16, 7, 40

1,2

The width pattern (A341969) of the symmetric representation of sigma for a number with k >= 1 odd divisors has length 2*k - 1.

a(p) = p for any prime number p is realized by the m+1 numbers 3^(p-1), ..., 2^m * 3^(p-1) which contain m+1-p duplicates, where m = floor(log_2(3^(p-1))). Each width pattern first increases to a level 1 <= i <= p and then alternates between i and i-1 up to the diagonal of the symmetric representation of sigma resulting in p distinct patterns.

For some numbers n = 2^m * q, q odd and not prime, that are the least instantiations of a width pattern their odd parts q may not be the least instantiations of a width pattern, examples are 78, 1014, 12246 and 171366 with 4, 6, 8 and 10 odd divisors, respectively (see row 2 of the table in A367377).

Conjecture: a(9) = 28.

The least number instantiating the 28th width pattern, 12345654345654321, is n = 43356672, found in a search up to 5*10^9.

Table of width pattern counts of the symmetric representation of sigma and of all possible symmetric patterns:

# odd divisors 1 2 3 4 5 6 7 8 9 10 11 12

pattern count 1 2 3 6 5 16 7 40 28? >=47 11 >=223

A001405 1 2 3 6 10 20 35 70 126 252 462 924

The 4 symmetric patterns 10123232101, 10123432101, 12101010121 and 12123432121 cannot be instantiated as width patterns of numbers with 6 odd divisors.

30 of the 70 possible symmetric patterns of numbers n = 2^m * q, m>=0 and q odd, with 8 odd divisors cannot be instantiated as width patterns of the symmetric representation of sigma(n) since their sequence of widths contradicts the order of the odd divisors d_i of n and of the numbers 2^(m+1) * d_i and the positions of their corresponding 1's in the rows of the triangle of widths in A249223.

allocated

nonn

Hartmut F. W. Hoft, Dec 05 2023

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allocated for Hartmut F. W. Hoft

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