A250071 -id:A250071 - cp's OEIS frontend

A053640 Number of divisors of highly composite odd numbers (A053624).

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 24, 32, 36, 40, 48, 54, 64, 72, 80, 96, 108, 120, 128, 144, 160, 192, 216, 240, 256, 288, 320, 384, 432, 480, 512, 576, 640, 720, 768, 864, 960, 1024, 1152, 1280, 1440, 1536, 1728, 1920, 2048, 2304, 2560, 2880, 3072, 3456

Offset: 1

Stefano Lanfranco (lastefano(AT)yahoo.it), Mar 22 2000

nonn

a(n) is the number of divisors of the odd parts of A340506(n); see also comments in A250071. - Hartmut F. W. Hoft, Mar 29 2022

			a(15) = 6 since A053624(6) = 45 has the 6 divisors 1, 3, 5, 9, 15, 45. - _Hartmut F. W. Hoft_, Mar 29 2022

Amiram Eldar, Table of n, a(n) for n = 1..363 (terms 1..170 from Ray Chandler)

Cf. A000005, A053624.

Cf. A237593, A237048, A237270, A250071, A340506.

a053640[n_] := Module[{recL={1}, q, d}, For[q=1, q<=n, q+=2, d=Length[Divisors[q]]; If[Last[recL]Hartmut F. W. Hoft, Mar 29 2022 *)

a(n) = A000005(A053624(n)).

Extended by Ray Chandler, Jan 12 2012

A340506 For those rows n of A249223 which are weakly increasing, let w(n) denote the maximal entry in the row: sequence gives values of n for which w(n) sets a new record.

Original entry on oeis.org

1, 6, 72, 120, 1440, 6720, 28800, 80640, 483840, 1612800, 5806080, 7096320, 85155840, 283852800, 510935040, 1476034560, 7947878400, 17712414720, 29520691200, 106274488320, 354248294400, 1653158707200, 2125489766400, 4817776803840, 8029628006400, 28906660823040

Offset: 1

N. J. A. Sloane, Jan 23 2021

This is a companion to A250071 (and is derived from the data for that sequence), which lists the first time k appears as a width.
The record values are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, but more data is needed to identify this sequence.
The odd part of a(n) is A053624(n), n>=1. The record values 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, ... are the beginning of A053640. - Hartmut F. W. Hoft, Mar 29 2022

			a(4) = 120 = 2^3 * A053624(4) = 2^3 * 15 and a(7) = 28800 = 2^7 * A053624(7) = 2^7 * 225. - _Hartmut F. W. Hoft_, Mar 29 2022

Cf. A237270, A237593, A249223, A250068, A250070, A250071.

Cf. A053624, A053640.

prevPower2[k_] := If[k==1, 1, 2^(Ceiling[Log[2, k]]-1)]
a340506[n_] := Module[{recL={{1, 1}}, q, d, pp}, For[q=1, q<=n, q+=2, d=DivisorSigma[0, q]; pp=prevPower2[q] q; If[First[Last[recL]]Hartmut F. W. Hoft, Mar 29 2022 *)

a(n) = 2^t(n) * A053624(n), n > 1, where t(n) is the largest exponent satisfying 2^t(n) < A053624(n) and A053624(n) is the odd part of a(n) - see the comment in A250071. - Hartmut F. W. Hoft, Mar 29 2022

a(12)-a(26) from Hartmut F. W. Hoft, Mar 29 2022

A244250 Triangle read by rows in which row n lists the widths in the first octant of the symmetric representation of sigma(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Omar E. Pol, Oct 26 2014

For the definition of k-th width of the symmetric representation of sigma(n) see A249351.
Row n list the first n terms of the n-th row of A249351.
It appears that the leading diagonal is also A067742 (which was conjectured by Michel Marcus in the entry A237593 and checked with two Mathematica functions up to n = 100000 by Hartmut F. W. Hoft).
For more information see A237591, A237593.

			Triangle begins:
1;
1, 1;
1, 1, 0;
1, 1, 1, 1;
1, 1, 1, 0, 0;
1, 1, 1, 1, 1, 2;
1, 1, 1, 1, 0, 0, 0;
1, 1, 1, 1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 0, 0, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 0;
1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0;
1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2;
1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0;
...

Cf. A000203, A067742, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A238443, A239660, A239932-A239934, A240542, A241008, A241010, A244360, A244361, A245685, A246955, A246956, A247687, A249223, A249351, A250068, A250070, A250071.

A338535 a(n) is the smallest number k for which the width n at the diagonal equals the maximum width of the symmetric representation of sigma(k).

Original entry on oeis.org

1, 6, 72, 120, 3528, 840, 3600, 2520, 28800, 10080, 129600, 36960, 259200, 50400, 176400, 83160, 352800, 138600

Offset: 1

Hartmut F. W. Hoft, Nov 01 2020

All numbers computed so far for this sequence have a symmetric representation of sigma that consists of a single region.

Additional values computed through 2000000 are a(20,21,22,24,26,30) = (277200, 1411200, 360360, 960960, 942480, 1884960).

			a(3) = 72 = 2^3*3^2 is in the sequence since it is the smallest with maximum width 3 and width 3 at the diagonal for its symmetric representation of sigma. Number 60 is the smallest number with maximum width 3, but it has width 2 at the diagonal (see also Ax22222). a(3) has the following pattern for the widths of the legs up to the diagonal (see A249223): 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3 for its single region.
a(5) = 3528 = 2^3*3^2*7^2 is in the sequence, but does not have a monotone increase of widths up to the diagonal in its single region: 1...2...3...4...3...4...3...4...5 (there are 83 legs to the diagonal).

Cf. A128605, A235791, A237048, A237591, A237593, A249223, A250071.

(* Functions row[] and a237048[] are defined in A237048 *)
widthQ0[n_] := Module[{r=row[n], cW=0, mW=0, k}, For[k=1, k<=r, k++, cW+=(-1)^(k+1) a237048[n, k]; If[cW>mW, mW=cW]]; If[mW==cW, cW, 0]]
a338535[n_, b_] := Module[{list=Table[0, {b}], k, wQ}, For[k=1, k<=n, k++, wQ=widthQ0[k]; If[wQ!=0&&list[[wQ]]==0, list[[wQ]]=k]]; list]
Take[a338535[500000,25],18] (* sequence data *)

A367370 a(k) is the number of different widths patterns in the symmetric representation of sigma for numbers having k odd divisors.

Original entry on oeis.org

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

Offset: 1

Hartmut F. W. Hoft, Dec 05 2023

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.

			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.

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

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 *)

A378470 a(n) is the smallest number k for which the width pattern of the symmetric representation of sigma(k), SRS(k), consists of two unimodal parts of maximum width n.

Original entry on oeis.org

3, 78, 10728, 28920, 53752896, 4157280, 278628512256, 90323520, 1658908800, 21499810560, 7487812494923563008, 13005699840, 155267279705546496147456, 111451576596480, 8599694054400, 468208581120, 4172630516011611848266349543424, 5202323481600, 21630916595004029113587563614961664, 67421367982080

Offset: 1

Hartmut F. W. Hoft, Nov 27 2024

nonn

Let the nonincreasing multiset cL = { c_1, ... , c_s } be a factorization of n, let dL = { d_1, ... , d_s } be any set of s distinct odd primes, let q = dL^(cL - 1) = d_1^(c_1 - 1) * ... * d_s^(c_s - 1), and let k satisfy 2^k < q < 2^(k+1). Then SRS(2^k * q) is unimodal of maximum height n, 2^k * q has 2n odd divisors and its width pattern has 2n-1 entries. The smallest possible choice for 2^k * q is with the increasing sequence of odd primes d_i = p_(i+1), 1 <= i <= s. The overall smallest 2^k * q is the minimum among all factorizations of n. The smallest number m for which SRS(m) has two unimodal parts of maximum width n requires the additional prime factor r > 2^(k+1) * q which yields m = 2^k * q * r.

This sequence is column 2 in the array of A367377 and a(2) = A370206(1).

			a(2) = 78  is in the sequence since SRS(78) consists of two parts with width pattern 1 2 1 0 1 2 1 and 78 is the smallest number with those properties.
a(3) = 10728 = 2^3 * 3^2 * 149 is in the sequence since SRS(10728) consists of two parts with width pattern 1 2 3 2 1 0 1 2 3 2 1 and 10728 is the smallest number with those properties.
a(6) = 4157280 = 2^5 * 3^2 * 5 * 2887 is in the sequence. The two factorizations of 6 are {6} and {3, 2} so that with 3^5 = 243 and 3^2 * 5^1 = 45 the inequality 2^5 < 45 < 2^6 determines the single unimodular SRS(32 * 45) of maximum width 6, A250071(6) = 1440. Since 2887 is the smallest prime exceeding 2^6 * 3^2 * 5, 4157280 is the smallest number with SRS(4157280) consisting of two unimodular parts of maximum width 6.

Cf. A003056, A082662, A162247, A235791, A237270, A237591, A237593, A239929, A250071, A341969, A370206.

(* function f[ ] by T. D. Noe in A162247 *)
sF[n_] := Min[Map[Apply[Times, Prime[Range[2, Length[#]+1]]^#]&, Map[Reverse[#-1]&, f[n]]]]
f2U[n_] := Module[{s=sF[n], k, p}, k=Floor[Log[2, s]]; p=NextPrime[2^(k+1) s]; 2^k s p]
a378470[n_] := Map[f2U, Range[n]]
a378470[20]

a(p) = 2^k * 3^(p-1) * r, for odd primes p, with 2^k < 3^(p-1) < 2^(k+1) and r > 2^(k+1) * 3^(p-1) least prime, i.e., k = floor( (p-1)*(log_2 (3)) ) and r = prime( primepi(2^(k+1) * 3^(p-1)) + 1 ).

cp's OEIS Frontend

A053640 Number of divisors of highly composite odd numbers (A053624).

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A340506 For those rows n of A249223 which are weakly increasing, let w(n) denote the maximal entry in the row: sequence gives values of n for which w(n) sets a new record.

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A244250 Triangle read by rows in which row n lists the widths in the first octant of the symmetric representation of sigma(n).

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A338535 a(n) is the smallest number k for which the width n at the diagonal equals the maximum width of the symmetric representation of sigma(k).

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A367370 a(k) is the number of different widths patterns in the symmetric representation of sigma for numbers having k odd divisors.

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A378470 a(n) is the smallest number k for which the width pattern of the symmetric representation of sigma(k), SRS(k), consists of two unimodal parts of maximum width n.

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