A167865 -id:A167865 - cp's OEIS frontend

A341450 Number of strict integer partitions of n that are empty or have smallest part not dividing all the others.

1, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 6, 3, 9, 9, 12, 12, 20, 18, 28, 27, 37, 42, 55, 51, 74, 80, 98, 105, 136, 137, 180, 189, 232, 255, 308, 320, 403, 434, 512, 551, 668, 706, 852, 915, 1067, 1170, 1370, 1453, 1722, 1860, 2145, 2332, 2701, 2899, 3355, 3626, 4144

Offset: 0

Gus Wiseman, Apr 15 2021

nonn

Alternative name: Number of strict integer partitions of n with no part dividing all the others.

			The a(0) = 1 through a(15) = 12 strict partitions (empty columns indicated by dots, 0 represents the empty partition, A..D = 10..13):
  0  .  .  .  .  32   .  43   53   54    64    65    75    76    86     87
                         52        72    73    74    543   85    95     96
                                   432   532   83    732   94    A4     B4
                                               92          A3    B3     D2
                                               542         B2    653    654
                                               632         643   743    753
                                                           652   752    762
                                                           742   932    843
                                                           832   5432   852
                                                                        942
                                                                        A32
                                                                        6432

The complement is counted by A097986 (non-strict: A083710, rank: A339563).
The complement with no 1's is A098965 (non-strict: A083711).
The non-strict version is A338470.
The Heinz numbers of these partitions are A339562 (non-strict: A342193).
The case with greatest part not divisible by all others is A343379.
The case with greatest part divisible by all others is A343380.
A000009 counts strict partitions (non-strict: A000041).
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A167865 counts strict chains of divisors > 1 summing to n.
Sequences with similar formulas: A024994, A047966, A047968, A168111.
Cf. A001787, A001792, A064410, A264401, A343342, A343381, A343382.

Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&!And@@IntegerQ/@(#/Min@@#)&]],{n,0,30}]

a(n > 0) = A000009(n) - Sum_{d|n} A025147(d-1).

A336500 Number of divisors d|n with distinct prime multiplicities such that the quotient n/d also has distinct prime multiplicities.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 4, 2, 2, 2, 5, 2, 4, 2, 4, 2, 2, 2, 6, 3, 2, 4, 4, 2, 0, 2, 6, 2, 2, 2, 6, 2, 2, 2, 6, 2, 0, 2, 4, 4, 2, 2, 8, 3, 4, 2, 4, 2, 6, 2, 6, 2, 2, 2, 4, 2, 2, 4, 7, 2, 0, 2, 4, 2, 0, 2, 8, 2, 2, 4, 4, 2, 0, 2, 8, 5, 2, 2, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Aug 06 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			The a(1) = 1 through a(16) = 5 divisors:
  1  1  1  1  1  2  1  1  1  2  1  1  1  2  3  1
     2  3  2  5  3  7  2  3  5 11  3 13  7  5  2
           4           4  9        4           4
                       8          12           8
                                              16

A336419 is the version for superprimorials.
A336568 gives positions of zeros.
A336869 is the restriction to factorials.
A007425 counts divisors of divisors.
A056924 counts divisors greater than their quotient.
A074206 counts chains of divisors from n to 1.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime multiplicities.
A336424 counts factorizations using A130091.
A336422 counts divisible pairs of divisors, both in A130091.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336423 counts chains in A130091, with maximal version A336569.
A336568 gives numbers not a product of two elements of A130091.
A336571 counts divisor sets using A130091, with maximal version A336570.
Cf. A000005, A001055, A002033, A098859, A124010, A167865, A253249, A336870.

Table[Length[Select[Divisors[n],UnsameQ@@Last/@FactorInteger[#]&&UnsameQ@@Last/@FactorInteger[n/#]&]],{n,25}]

A338470 Number of integer partitions of n with no part dividing all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 3, 2, 5, 5, 13, 7, 23, 21, 33, 35, 65, 55, 104, 97, 151, 166, 252, 235, 377, 399, 549, 591, 846, 858, 1237, 1311, 1749, 1934, 2556, 2705, 3659, 3991, 5090, 5608, 7244, 7841, 10086, 11075, 13794, 15420, 19195, 21003, 26240, 29089, 35483

Offset: 0

Gus Wiseman, Mar 23 2021

nonn

Alternative name: Number of integer partitions of n that are empty or have smallest part not dividing all the others.

			The a(5) = 1 through a(12) = 7 partitions (empty column indicated by dot):
  (32)  .  (43)   (53)   (54)    (64)    (65)     (75)
           (52)   (332)  (72)    (73)    (74)     (543)
           (322)         (432)   (433)   (83)     (552)
                         (522)   (532)   (92)     (732)
                         (3222)  (3322)  (443)    (4332)
                                         (533)    (5322)
                                         (542)    (33222)
                                         (632)
                                         (722)
                                         (3332)
                                         (4322)
                                         (5222)
                                         (32222)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The complement is A083710 (strict: A097986).
The strict case is A341450.
The Heinz numbers of these partitions are A342193.
The dual version is A343341.
The case with maximum part not divisible by all the others is A343342.
The case with maximum part divisible by all the others is A343344.
A000005 counts divisors.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A001787 count normal multisets with a selected position.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A167865 counts strict chains of divisors > 1 summing to n.
A276024 counts positive subset sums.
Sequences with similar formulas: A024994, A047966, A047968, A168111.
Cf. A001792, A064391, A064410, A066186, A067824, A083711, A098965, A264401, A339562, A339563.

Table[Length[Select[IntegerPartitions[n],#=={}||!And@@IntegerQ/@(#/Min@@#)&]],{n,0,30}]
(* Second program: *)
a[n_] := If[n == 0, 1, PartitionsP[n] - Sum[PartitionsP[d-1], {d, Divisors[n]}]];
a /@ Range[0, 50] (* Jean-François Alcover, May 09 2021, after Andrew Howroyd *)

a(n)={numbpart(n) - if(n, sumdiv(n, d, numbpart(d-1)))} \\ Andrew Howroyd, Mar 25 2021

a(n) = A000041(n) - Sum_{d|n} A000041(d-1) for n > 0. - Andrew Howroyd, Mar 25 2021

A342083 Number of chains of strictly inferior divisors from n to 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 4, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 6, 1, 3, 2, 3, 1, 5, 2, 4, 2, 2, 1, 7, 1, 2, 3, 3, 2, 5, 1, 3, 2, 4, 1, 8, 1, 2, 3, 3, 2, 5, 1, 6, 2, 2, 1, 7, 2, 2, 2

Offset: 1

Gus Wiseman, Feb 28 2021

nonn

We define a divisor d|n to be strictly inferior if d < n/d. Strictly inferior divisors are counted by A056924 and listed by A341674.
These chains have first-quotients (in analogy with first-differences) that are term-wise > their decapitation (maximum element removed). Equivalently, x > y^2 for all adjacent x, y. For example, the divisor chain q = 60/6/2/1 has first-quotients (10,3,2), which are > (6,2,1), so q is counted under a(60).
Also the number of factorizations of n where each factor is greater than the product of all previous factors.

			The a(n) chains for n = 2, 6, 12, 24, 42, 48, 60, 72:
  2/1  6/1    12/1    24/1    42/1      48/1      60/1      72/1
       6/2/1  12/2/1  24/2/1  42/2/1    48/2/1    60/2/1    72/2/1
              12/3/1  24/3/1  42/3/1    48/3/1    60/3/1    72/3/1
                      24/4/1  42/6/1    48/4/1    60/4/1    72/4/1
                              42/6/2/1  48/6/1    60/5/1    72/6/1
                                        48/6/2/1  60/6/1    72/8/1
                                                  60/6/2/1  72/6/2/1
                                                            72/8/2/1
The a(n) factorizations for n = 2, 6, 12, 24, 42, 48, 60, 72:
  2  6    12   24    42     48     60      72
     2*3  2*6  3*8   6*7    6*8    2*30    8*9
          3*4  4*6   2*21   2*24   3*20    2*36
               2*12  3*14   3*16   4*15    3*24
                     2*3*7  4*12   5*12    4*18
                            2*3*8  6*10    6*12
                                   2*3*10  2*4*9
                                           2*3*12

Amiram Eldar, Table of n, a(n) for n = 1..10000

The restriction to powers of 2 is A040039.
Not requiring strict inferiority gives A074206 (ordered factorizations).
The weakly inferior version is A337135.
The strictly superior version is A342084.
The weakly superior version is A342085.
The additive version is A342098, or A000929 allowing equality.
A000005 counts divisors.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n-1, with strict case A122651.
A038548 counts inferior (or superior) divisors.
A056924 counts strictly inferior (or strictly superior) divisors.
A067824 counts strict chains of divisors starting with n.
A167865 counts strict chains of divisors > 1 summing to n.
A207375 lists central divisors.
A253249 counts strict chains of divisors.
A334996 counts ordered factorizations by product and length.
A334997 counts chains of divisors of n by length.
A342086 counts chains of divisors with strictly increasing quotients > 1.
- Inferior: A033676, A066839, A072499, A161906.
- Superior: A033677, A070038, A161908.
- Strictly Inferior: A060775, A070039, A333806, A341674.
- Strictly Superior: A048098, A064052, A140271, A238535, A341673.
Cf. A000203, A001248, A002033, A006530, A018819, A020639, A045690, A337105, A342087, A342094, A342095, A342096, A342097.

cen[n_]:=If[n==1,{{1}},Prepend[#,n]&/@Join@@cen/@Select[Divisors[n],#

G.f.: x + Sum_{k>=1} a(k) * x^(k*(k + 1)) / (1 - x^k). - Ilya Gutkovskiy, Nov 03 2021

A342084 Number of chains of distinct strictly superior divisors starting with n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 9, 1, 2, 2, 4, 1, 7, 1, 6, 2, 2, 2, 10, 1, 2, 2, 9, 1, 6, 1, 4, 4, 2, 1, 19, 1, 4, 2, 4, 1, 8, 2, 9, 2, 2, 1, 20, 1, 2, 4, 10, 2, 6, 1, 4, 2, 7, 1, 29, 1, 2, 4, 4, 2, 6, 1, 19, 3, 2, 1, 19, 2

Offset: 1

Gus Wiseman, Feb 28 2021

nonn

We define a divisor d|n to be strictly superior if d > n/d. Strictly superior divisors are counted by A056924 and listed by A341673.
These chains have first-quotients (in analogy with first-differences) that are term-wise < their decapitation (maximum element removed). Equivalently, x < y^2 for all adjacent x, y. For example, the divisor chain q = 30/6/3 has first-quotients (5,2), which are < (6,3), so q is counted under a(30).
Also the number of ordered factorizations of n where each factor is less than the product of all previous factors.

			The a(n) chains for n = 2, 6, 12, 16, 24, 30, 32, 36:
  2  6    12      16      24         30       32         36
     6/3  12/4    16/8    24/6       30/6     32/8       36/9
          12/6    16/8/4  24/8       30/10    32/16      36/12
          12/6/3          24/12      30/15    32/8/4     36/18
                          24/6/3     30/6/3   32/16/8    36/12/4
                          24/8/4     30/10/5  32/16/8/4  36/12/6
                          24/12/4    30/15/5             36/18/6
                          24/12/6                        36/18/9
                          24/12/6/3                      36/12/6/3
                                                         36/18/6/3
The a(n) ordered factorizations for n = 2, 6, 12, 16, 24, 30, 32, 36:
  2  6    12     16     24       30     32       36
     3*2  4*3    8*2    6*4      6*5    8*4      9*4
          6*2    4*2*2  8*3      10*3   16*2     12*3
          3*2*2         12*2     15*2   4*2*4    18*2
                        3*2*4    3*2*5  8*2*2    4*3*3
                        4*2*3    5*2*3  4*2*2*2  6*2*3
                        4*3*2    5*3*2           6*3*2
                        6*2*2                    9*2*2
                        3*2*2*2                  3*2*2*3
                                                 3*2*3*2

Amiram Eldar, Table of n, a(n) for n = 1..10000

The restriction to powers of 2 is A045690, with reciprocal version A040039.
The inferior version is A337135.
The strictly inferior version is A342083.
The superior version is A342085.
The additive version allowing equality is A342094 or A342095.
The additive version is A342096 or A342097.
A000005 counts divisors.
A001055 counts factorizations.
A003238 counts divisibility chains summing to n-1, with strict case A122651.
A038548 counts inferior (or superior) divisors.
A056924 counts strictly inferior (or strictly superior) divisors.
A067824 counts strict chains of divisors starting with n.
A074206 counts strict chains of divisors from n to 1 (also ordered factorizations).
A167865 counts strict chains of divisors > 1 summing to n.
A207375 lists central divisors.
A253249 counts strict chains of divisors.
A334996 counts ordered factorizations by product and length.
A334997 counts chains of divisors of n by length.
- Inferior: A033676, A063962, A066839, A072499, A161906.
- Superior: A033677, A070038, A161908, A341591.
- Strictly Inferior: A060775, A070039, A333806, A341674.
- Strictly Superior: A064052/A048098, A140271, A238535, A341642, A341673.
Cf. A000203, A000929, A001248, A006530, A018819, A020639, A169594, A337105, A342098.

ceo[n_]:=Prepend[Prepend[#,n]&/@Join@@ceo/@Select[Most[Divisors[n]],#>n/#&],{n}];
Table[Length[ceo[n]],{n,100}]

a(2^n) = A045690(n).

A336568 Numbers that are not a product of two numbers each having distinct prime multiplicities.

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 210, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 330, 345, 354, 357, 366, 370, 374, 385, 390, 399, 402, 406, 410, 418, 420, 426, 429

Offset: 1

Gus Wiseman, Aug 06 2020

nonn

First differs from A007304 and A093599 in having 210.
First differs from A287483 in having 222.
First differs from A350352 in having 420.
A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			Selected terms together with their prime indices:
   660: {1,1,2,3,5}
   798: {1,2,4,8}
   840: {1,1,1,2,3,4}
  3120: {1,1,1,1,2,3,6}
  9900: {1,1,2,2,3,3,5}

A336500 has zeros at these positions.
A007425 counts divisors of divisors.
A056924 counts divisors greater than their quotient.
A074206 counts strict chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A336424 counts factorizations using A130091.
A336422 counts divisible pairs of divisors, both in A130091.
A327498 is the maximum divisor with distinct prime multiplicities.
A336423 counts chains in A130091, with maximal version A336569.
A336571 counts divisor sets using A130091, with maximal version A336570.
Cf. A001055, A002033, A098859, A124010, A167865, A253249, A336420.

strsig[n_]:=UnsameQ@@Last/@FactorInteger[n]
Select[Range[100],Function[n,Select[Divisors[n],strsig[#]&&strsig[n/#]&]=={}]]

A336423 Number of strict chains of divisors from n to 1 using terms of A130091 (numbers with distinct prime multiplicities).

Original entry on oeis.org

1, 1, 1, 2, 1, 0, 1, 4, 2, 0, 1, 5, 1, 0, 0, 8, 1, 5, 1, 5, 0, 0, 1, 14, 2, 0, 4, 5, 1, 0, 1, 16, 0, 0, 0, 0, 1, 0, 0, 14, 1, 0, 1, 5, 5, 0, 1, 36, 2, 5, 0, 5, 1, 14, 0, 14, 0, 0, 1, 0, 1, 0, 5, 32, 0, 0, 1, 5, 0, 0, 1, 35, 1, 0, 5, 5, 0, 0, 1, 36, 8, 0, 1, 0

Offset: 1

Gus Wiseman, Jul 27 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			The a(n) chains for n = 4, 8, 12, 16, 24, 32:
  4/1    8/1      12/1      16/1        24/1         32/1
  4/2/1  8/2/1    12/2/1    16/2/1      24/2/1       32/2/1
         8/4/1    12/3/1    16/4/1      24/3/1       32/4/1
         8/4/2/1  12/4/1    16/8/1      24/4/1       32/8/1
                  12/4/2/1  16/4/2/1    24/8/1       32/16/1
                            16/8/2/1    24/12/1      32/4/2/1
                            16/8/4/1    24/4/2/1     32/8/2/1
                            16/8/4/2/1  24/8/2/1     32/8/4/1
                                        24/8/4/1     32/16/2/1
                                        24/12/2/1    32/16/4/1
                                        24/12/3/1    32/16/8/1
                                        24/12/4/1    32/8/4/2/1
                                        24/8/4/2/1   32/16/4/2/1
                                        24/12/4/2/1  32/16/8/2/1
                                                     32/16/8/4/1
                                                     32/16/8/4/2/1

A336569 is the maximal case.
A336571 does not require n itself to have distinct prime multiplicities.
A000005 counts divisors.
A007425 counts divisors of divisors.
A074206 counts strict chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts nonempty strict chains of divisors.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336422 counts divisible pairs of divisors, both in A130091.
A336424 counts factorizations using A130091.
A336500 counts divisors of n in A130091 with quotient also in A130091.
A337256 counts strict chains of divisors.
Cf. A001055, A002033, A005117, A032741, A067824, A071625, A118914, A124010, A167865, A336568, A336570, A336941.

strchns[n_]:=If[n==1,1,If[!UnsameQ@@Last/@FactorInteger[n],0,Sum[strchns[d],{d,Select[Most[Divisors[n]],UnsameQ@@Last/@FactorInteger[#]&]}]]];
Table[strchns[n],{n,100}]

A337105 Number of strict chains of divisors from n! to 1.

Original entry on oeis.org

1, 1, 1, 3, 20, 132, 1888, 20128, 584000, 17102016, 553895936, 11616690176, 743337949184, 19467186157568, 999551845713920, 66437400489711616, 10253161206302064640, 388089999627661557760, 53727789519052432998400, 2325767421950553303285760, 365546030278816140131041280

Offset: 0

Gus Wiseman, Aug 17 2020

nonn

			The a(4) = 20 chains:
  24/1  24/2/1   24/4/2/1   24/8/4/2/1
        24/3/1   24/6/2/1   24/12/4/2/1
        24/4/1   24/6/3/1   24/12/6/2/1
        24/6/1   24/8/2/1   24/12/6/3/1
        24/8/1   24/8/4/1
        24/12/1  24/12/2/1
                 24/12/3/1
                 24/12/4/1
                 24/12/6/1

Andrew Howroyd, Table of n, a(n) for n = 0..200

A325617 is the maximal case.
A336941 is the version for superprimorials.
A337104 counts the case with distinct prime multiplicities.
A337071 is the case not necessarily ending with 1.
A000005 counts divisors.
A000142 lists factorial numbers.
A001055 counts factorizations.
A027423 counts divisors of factorial numbers.
A067824 counts chains of divisors starting with n.
A074206 counts chains of divisors from n to 1.
A076716 counts factorizations of factorial numbers.
A253249 counts chains of divisors.
A336423 counts chains using A130091, with maximal case A336569.
A336942 counts chains using A130091 from A006939(n) to 1.
Cf. A001013, A001055, A022559, A124010, A167865, A337070, A337074.

b:= proc(n) option remember; 1 +
      add(b(d), d=numtheory[divisors](n) minus {n})
    end:
a:= n-> ceil(b(n!)/2):
seq(a(n), n=0..14);  # Alois P. Heinz, Aug 23 2020

chnsc[n_]:=Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,DeleteCases[Divisors[n],1|n]}],{n}];
Table[Length[chnsc[n!]],{n,0,5}]

a(n) = A337071(n)/2 for n > 1.

a(n) = A074206(n!).

a(19)-a(20) from Alois P. Heinz, Aug 22 2020

A343382 Number of strict integer partitions of n with either (1) no part dividing all the others or (2) no part divisible by all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 9, 9, 13, 18, 21, 26, 34, 38, 48, 57, 67, 81, 99, 110, 133, 157, 183, 211, 250, 282, 330, 380, 437, 502, 575, 648, 748, 852, 967, 1095, 1250, 1405, 1597, 1801, 2029, 2287, 2579, 2883, 3245, 3638, 4077, 4557, 5107, 5691, 6356

Offset: 0

Gus Wiseman, Apr 16 2021

nonn

Alternative name: Number of strict integer partitions of n that are either (1) empty, or (2) have smallest part not dividing all the others, or (3) have greatest part not divisible by all the others.

			The a(0) = 1 through a(11) = 9 partitions (empty columns indicated by dots):
  ()  .  .  .  .  (3,2)  (3,2,1)  (4,3)  (5,3)    (5,4)    (6,4)      (6,5)
                                  (5,2)  (4,3,1)  (7,2)    (7,3)      (7,4)
                                         (5,2,1)  (4,3,2)  (5,3,2)    (8,3)
                                                  (5,3,1)  (5,4,1)    (9,2)
                                                           (7,2,1)    (5,4,2)
                                                           (4,3,2,1)  (6,3,2)
                                                                      (6,4,1)
                                                                      (7,3,1)
                                                                      (5,3,2,1)

The first condition alone gives A341450.
The non-strict version is A343346 (Heinz numbers: A343343).
The second condition alone gives A343377.
The strict complement is A343378.
The version for "and" instead of "or" is A343379.
A000005 counts divisors.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A083710, A097986, A130689, A200745, A264401, A338470, A339562, A342193, A343337, A343338, A343341, A343342.

Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&!And@@IntegerQ/@(#/Min@@#)||UnsameQ@@#&&!And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]

A337135 a(1) = 1; for n > 1, a(n) = Sum_{d|n, d <= sqrt(n)} a(d).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 4, 2, 2, 1, 5, 2, 2, 2, 4, 1, 4, 1, 4, 2, 2, 2, 7, 1, 2, 2, 5, 1, 5, 1, 4, 3, 2, 1, 7, 2, 3, 2, 4, 1, 5, 2, 5, 2, 2, 1, 8, 1, 2, 3, 6, 2, 5, 1, 4, 2, 4, 1, 9, 1, 2, 3, 4, 2, 5, 1, 7, 4, 2, 1, 8, 2, 2, 2, 6, 1, 8, 2, 4, 2, 2, 2

Offset: 1

Ilya Gutkovskiy, Nov 21 2020

nonn

From Gus Wiseman, Mar 05 2021: (Start)
This sequence counts all of the following essentially equivalent things:
1. Chains of distinct inferior divisors from n to 1, where a divisor d|n is inferior if d <= n/d. Inferior divisors are counted by A038548 and listed by A161906.
2. Chains of divisors from n to 1 whose first-quotients (in analogy with first-differences) are term-wise greater than or equal to their decapitation (maximum element removed). For example, the divisor chain q = 60/4/2/1 has first-quotients (15,2,2), which are >= (4,2,1), so q is counted under a(60).
3. Chains of divisors from n to 1 such that x >= y^2 for all adjacent x, y.
4. Factorizations of n where each factor is greater than or equal to the product of all previous factors.
(End)

			From _Gus Wiseman_, Mar 05 2021: (Start)
The a(n) chains for n = 1, 2, 4, 12, 16, 24, 36, 60:
  1  2/1  4/1    12/1    16/1      24/1      36/1      60/1
          4/2/1  12/2/1  16/2/1    24/2/1    36/2/1    60/2/1
                 12/3/1  16/4/1    24/3/1    36/3/1    60/3/1
                         16/4/2/1  24/4/1    36/4/1    60/4/1
                                   24/4/2/1  36/6/1    60/5/1
                                             36/4/2/1  60/6/1
                                             36/6/2/1  60/4/2/1
                                                       60/6/2/1
The a(n) factorizations for n = 2, 4, 12, 16, 24, 36, 60:
    2  4    12   16     24     36     60
       2*2  2*6  2*8    3*8    4*9    2*30
            3*4  4*4    4*6    6*6    3*20
                 2*2*4  2*12   2*18   4*15
                        2*2*6  3*12   5*12
                               2*2*9  6*10
                               2*3*6  2*2*15
                                      2*3*10
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..65536

Cf. A002033, A008578 (positions of 1's), A068108.
The restriction to powers of 2 is A018819.
Not requiring inferiority gives A074206 (ordered factorizations).
The strictly inferior version is A342083.
The strictly superior version is A342084.
The weakly superior version is A342085.
The additive version is A000929, or A342098 forbidding equality.
A000005 counts divisors, with sum A000203.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n-1, with strict case A122651.
A038548 counts inferior (or superior) divisors.
A056924 counts strictly inferior (or strictly superior) divisors.
A067824 counts strict chains of divisors starting with n.
A167865 counts strict chains of divisors > 1 summing to n.
A207375 lists central divisors.
A253249 counts strict chains of divisors.
A334996 counts ordered factorizations by product and length.
A334997 counts chains of divisors of n by length.
A342086 counts strict factorizations of divisors.
- Inferior: A033676, A066839, A072499, A161906.
- Superior: A033677, A070038, A161908.
- Strictly Inferior: A060775, A070039, A333806, A341674.
- Strictly Superior: A048098, A064052, A140271, A238535, A341673.
Cf. A001248, A006530, A020639, A040039, A045690, A337105, A342087, A342094, A342095, A342096, A342097.

a:= proc(n) option remember; `if`(n=1, 1, add(
      `if`(d<=n/d, a(d), 0), d=numtheory[divisors](n)))
    end:
seq(a(n), n=1..128);  # Alois P. Heinz, Jun 24 2021

a[1] = 1; a[n_] := a[n] = DivisorSum[n, a[#] &, # <= Sqrt[n] &]; Table[a[n], {n, 95}]
(* second program *)
asc[n_]:=Prepend[#,n]&/@Prepend[Join@@Table[asc[d],{d,Select[Divisors[n],#Gus Wiseman, Mar 05 2021 *)

G.f.: Sum_{k>=1} a(k) * x^(k^2) / (1 - x^k).

a(2^n) = A018819(n). - Gus Wiseman, Mar 08 2021

cp's OEIS Frontend

A341450 Number of strict integer partitions of n that are empty or have smallest part not dividing all the others.

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A336500 Number of divisors d|n with distinct prime multiplicities such that the quotient n/d also has distinct prime multiplicities.

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A338470 Number of integer partitions of n with no part dividing all the others.

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A342083 Number of chains of strictly inferior divisors from n to 1.

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A342084 Number of chains of distinct strictly superior divisors starting with n.

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A336568 Numbers that are not a product of two numbers each having distinct prime multiplicities.

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A336423 Number of strict chains of divisors from n to 1 using terms of A130091 (numbers with distinct prime multiplicities).

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A337105 Number of strict chains of divisors from n! to 1.

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