A339563 -id:A339563 - cp's OEIS frontend

A083710 Number of integer partitions of n with a part dividing all the other parts.

1, 1, 2, 3, 5, 6, 11, 12, 20, 25, 37, 43, 70, 78, 114, 143, 196, 232, 330, 386, 530, 641, 836, 1003, 1340, 1581, 2037, 2461, 3127, 3719, 4746, 5605, 7038, 8394, 10376, 12327, 15272, 17978, 22024, 26095, 31730, 37339, 45333, 53175, 64100, 75340, 90138

Offset: 0

N. J. A. Sloane, Jun 16 2003

Since the summand (part) which divides all the other summands is necessarily the smallest, an equivalent definition is: "Number of partitions of n such that smallest part divides every part." - Joerg Arndt, Jun 08 2009
The first few partitions that fail the criterion are 5=3+2, 7=5+2=4+3=3+2+2. So a(5) = A000041(5) - 1 = 6, a(7) = A000041(7) - 3 = 12. - Vladeta Jovovic, Jun 17 2003
Starting with offset 1 = inverse Mobius transform (A051731) of the partition numbers, A000041. - Gary W. Adamson, Jun 08 2009

			From _Gus Wiseman_, Apr 18 2021: (Start)
The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (41)     (33)      (61)
             (111)  (31)    (221)    (42)      (331)
                    (211)   (311)    (51)      (421)
                    (1111)  (2111)   (222)     (511)
                            (11111)  (321)     (2221)
                                     (411)     (3211)
                                     (2211)    (4111)
                                     (3111)    (22111)
                                     (21111)   (31111)
                                     (111111)  (211111)
                                               (1111111)
(End)

L. M. Chawla, M. O. Levan and J. E. Maxfield, On a restricted partition function and its tables, J. Natur. Sci. and Math., 12 (1972), 95-101.

Cf. A018783, A137587.
Cf. A000041, A051731. - Gary W. Adamson, Jun 08 2009
The case with no 1's is A083711.
The strict case is A097986.
The version for "divisible by" instead of "dividing" is A130689.
The case where there is also a part divisible by all the others is A130714.
The complement of these partitions is counted by A338470.
The Heinz numbers of these partitions are dense, complement of A342193.
The case where there is also no part divisible by all the others is A343345.
A000005 counts divisors.
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.
Cf. A001792, A098965, A264401, A339563, A343340, A343341, A343378.

with(combinat): with(numtheory): a := proc(n) c := 0: l := sort(convert(divisors(n), list)): for i from 1 to nops(l)-0 do c := c+numbpart(l[i]-1) od: RETURN(c): end: for j from 0 to 60 do printf(`%d, `, a(j)) od: # Zerinvary Lajos, Apr 14 2007

Table[Length[Select[IntegerPartitions[n],And@@IntegerQ/@(#/Min@@#)&]],{n,0,30}] (* Gus Wiseman, Apr 18 2021 *)

Equals left border of triangle A137587 starting (1, 2, 3, 5, 6, 11, ...). - Gary W. Adamson, Jan 27 2008
G.f.: 1 + Sum_{n>=1} x^n/eta(x^n). The g.f. for partitions into parts that are a multiple of n is x^n/eta(x^n), now sum over n. - Joerg Arndt, Jun 08 2009
Gary W. Adamson's comment is equivalent to the formula a(n) = Sum_{d|n} p(d-1) where p(i) = number of partitions of i (A000041(i)). Hence A083710 has g.f. Sum_{d>=1} p(d-1)*x^d/(1-x^d), - N. J. A. Sloane, Jun 08 2009

More terms from Vladeta Jovovic, Jun 17 2003

Name shortened by Gus Wiseman, Apr 18 2021

A097986 Number of strict integer partitions of n with a part dividing all the other parts.

Original entry on oeis.org

1, 1, 2, 2, 2, 4, 3, 5, 5, 7, 6, 12, 9, 13, 15, 20, 18, 28, 26, 37, 39, 47, 49, 71, 68, 85, 94, 117, 120, 159, 160, 201, 216, 257, 277, 348, 357, 430, 470, 562, 592, 720, 758, 901, 981, 1134, 1220, 1457, 1542, 1798, 1952, 2250, 2419, 2819, 3023, 3482, 3773, 4291

Offset: 1

Vladeta Jovovic, Oct 23 2004

If n > 0, we can assume such a part is the smallest. - Gus Wiseman, Apr 23 2021

Also the number of uniform (constant multiplicity) partitions of n containing 1, ranked by A367586. The strict case is A096765. The version without 1 is A329436. - Gus Wiseman, Dec 01 2023

			From _Gus Wiseman_, Dec 01 2023: (Start)
The a(1) = 1 through a(8) = 5 strict partitions with a part dividing all the other parts:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)
            (2,1)  (3,1)  (4,1)  (4,2)    (6,1)    (6,2)
                                 (5,1)    (4,2,1)  (7,1)
                                 (3,2,1)           (4,3,1)
                                                   (5,2,1)
The a(1) = 1 through a(8) = 5 uniform partitions containing 1:
  (1)  (11)  (21)   (31)    (41)     (51)      (61)       (71)
             (111)  (1111)  (11111)  (321)     (421)      (431)
                                     (2211)    (1111111)  (521)
                                     (111111)             (3311)
                                                          (11111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 500 terms from John Tyler Rascoe)

The non-strict version is A083710.
The case with no 1's is A098965.
The Heinz numbers of these partitions are A339563.
The strict complement is counted by A341450.
The version for "divisible by" instead of "dividing" is A343347.
The case where there is also a part divisible by all the others is A343378.
The case where there is no part divisible by all the others is A343381.
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.
Cf. A083711, A098743, A130689, A200745, A264401, A338470, A343377, A343379, A343380.
Cf. A023645, A025147, A047966, A072774, A096765, A329436, A367586.

Take[ CoefficientList[ Expand[ Sum[x^k*Product[1 + x^(k*i), {i, 2, 62}], {k, 62}]], x], {2, 60}] (* Robert G. Wilson v, Nov 01 2004 *)
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Or@@Table[And@@IntegerQ/@(#/x), {x,#}]&]], {n,0,30}] (* Gus Wiseman, Apr 23 2021 *)

A_x(N) = {my(x='x+O('x^N)); Vec(sum(k=1,N,x^k*prod(i=2,N-k, (1+x^(k*i)))))}
A_x(50) \\ John Tyler Rascoe, Nov 19 2024

a(n) = Sum_{d|n} A025147(d-1).
G.f.: Sum_{k>=1} (x^k*Product_{i>=2} (1+x^(k*i))).
a(n) ~ exp(Pi*sqrt(n/3)) / (8*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jul 06 2025

More terms from Robert G. Wilson v, Nov 01 2004

Name shortened by Gus Wiseman, Apr 23 2021

A339561 Products of distinct squarefree semiprimes.

Original entry on oeis.org

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 126, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159, 161, 166

Offset: 1

Gus Wiseman, Dec 13 2020

nonn

First differs from A320911 in lacking 36.
A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct strict pairs (a set of edges);
(2) n can be factored into distinct squarefree semiprimes;
(3) the prime signature of n is graphical.

			The sequence of terms together with their prime indices begins:
      1: {}        55: {3,5}         91: {4,6}
      6: {1,2}     57: {2,8}         93: {2,11}
     10: {1,3}     58: {1,10}        94: {1,15}
     14: {1,4}     60: {1,1,2,3}     95: {3,8}
     15: {2,3}     62: {1,11}       106: {1,16}
     21: {2,4}     65: {3,6}        111: {2,12}
     22: {1,5}     69: {2,9}        115: {3,9}
     26: {1,6}     74: {1,12}       118: {1,17}
     33: {2,5}     77: {4,5}        119: {4,7}
     34: {1,7}     82: {1,13}       122: {1,18}
     35: {3,4}     84: {1,1,2,4}    123: {2,13}
     38: {1,8}     85: {3,7}        126: {1,2,2,4}
     39: {2,6}     86: {1,14}       129: {2,14}
     46: {1,9}     87: {2,10}       132: {1,1,2,5}
     51: {2,7}     90: {1,2,2,3}    133: {4,8}
For example, the number 1260 can be factored into distinct squarefree semiprimes in two ways, (6*10*21) or (6*14*15), so 1260 is in the sequence. The number 69300 can be factored into distinct squarefree semiprimes in seven ways:
  (6*10*15*77)
  (6*10*21*55)
  (6*10*33*35)
  (6*14*15*55)
  (6*15*22*35)
  (10*14*15*33)
  (10*15*21*22),
so 69300 is in the sequence. A complete list of all strict factorizations of 24 is: (2*3*4), (2*12), (3*8), (4*6), (24), all of which contain at least one number that is not a squarefree semiprime, so 24 is not in the sequence.

Eric Weisstein's World of Mathematics, Graphical partition.

A309356 is a kind of universal embedding.
A320894 is the complement in A028260.
A320911 lists all (not just distinct) products of squarefree semiprimes.
A339560 counts the partitions with these Heinz numbers.
A339661 has nonzero terms at these positions.
A001358 lists semiprimes, with squarefree case A006881.
A005117 lists squarefree numbers.
A320656 counts factorizations into squarefree semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A058696 counts partitions of 2n (A300061).
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A209816 counts multigraphical partitions (A320924).
- A320921 counts connected graphical partitions (A320923).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339656 counts loop-graphical partitions (A339658).
- A339617 counts non-graphical partitions of 2n (A339618).
- A000569 counts graphical partitions (A320922).
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561 [this sequence]).
Cf. A001055, A001221, A002100, A007717, A030229, A112798, A320655, A320893, A338899, A338903, A339563, A339659.

sqs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqs[n/d],Min@@#>d&]],{d,Select[Divisors[n],SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
Select[Range[100],sqs[#]!={}&]

A339561 \/ A320894 = A028260.

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

Original entry on oeis.org

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).

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

A338916 Number of integer partitions of n that can be partitioned into distinct pairs of (possibly equal) parts.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 5, 6, 8, 12, 16, 21, 28, 37, 49, 64, 80, 104, 135, 169, 216, 268, 341, 420, 527, 654, 809, 991, 1218, 1488, 1828, 2213, 2687, 3262, 3934, 4754, 5702, 6849, 8200, 9819, 11693, 13937, 16562, 19659, 23262, 27577, 32493, 38341, 45112, 53059, 62265

Offset: 0

Gus Wiseman, Dec 10 2020

nonn

The multiplicities of such a partition form a loop-graphical partition (A339656, A339658).

			The a(2) = 1 through a(10) = 16 partitions:
  (11)  (21)  (22)  (32)    (33)    (43)    (44)    (54)      (55)
              (31)  (41)    (42)    (52)    (53)    (63)      (64)
                    (2111)  (51)    (61)    (62)    (72)      (73)
                            (2211)  (2221)  (71)    (81)      (82)
                            (3111)  (3211)  (3221)  (3222)    (91)
                                    (4111)  (3311)  (3321)    (3322)
                                            (4211)  (4221)    (3331)
                                            (5111)  (4311)    (4222)
                                                    (5211)    (4321)
                                                    (6111)    (4411)
                                                    (222111)  (5221)
                                                    (321111)  (5311)
                                                              (6211)
                                                              (7111)
                                                              (322111)
                                                              (421111)
For example, the partition (4,2,1,1,1,1) can be partitioned into {{1,1},{1,2},{1,4}}, and thus is counted under a(10).

Eric Weisstein's World of Mathematics, Graphical partition.

A320912 gives the Heinz numbers of these partitions.
A338915 counts the complement in even-length partitions.
A339563 counts factorizations of the same type.
A000070 counts non-multigraphical partitions of 2n, ranked by A339620.
A000569 counts graphical partitions, ranked by A320922.
A001358 lists semiprimes, with squarefree case A006881.
A058696 counts partitions of even numbers, ranked by A300061.
A209816 counts multigraphical partitions, ranked by A320924.
A320655 counts factorizations into semiprimes.
A322353 counts factorizations into distinct semiprimes.
A339617 counts non-graphical partitions of 2n, ranked by A339618.
A339655 counts non-loop-graphical partitions of 2n, ranked by A339657.
A339656 counts loop-graphical partitions, ranked by A339658.
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001055, A007717, A025065, A320656, A320732, A320893, A320921, A338898, A338902, A339564.

stfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[stfs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];
Table[Length[Select[IntegerPartitions[n],stfs[Times@@Prime/@#]!={}&]],{n,0,20}]

A027187(n) = a(n) + A338915(n).

More terms from Jinyuan Wang, Feb 14 2025

A342193 Numbers with no prime index dividing all the other prime indices.

Original entry on oeis.org

1, 15, 33, 35, 45, 51, 55, 69, 75, 77, 85, 91, 93, 95, 99, 105, 119, 123, 135, 141, 143, 145, 153, 155, 161, 165, 175, 177, 187, 195, 201, 203, 205, 207, 209, 215, 217, 219, 221, 225, 231, 245, 247, 249, 253, 255, 265, 275, 279, 285, 287, 291, 295, 297, 299

Offset: 1

Gus Wiseman, Apr 11 2021

nonn

Alternative name: 1 and numbers with smallest prime index not dividing all the other prime indices.
First differs from A339562 in having 45.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Also 1 and Heinz numbers of integer partitions with smallest part not dividing all the others (counted by A338470). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      1: {}         105: {2,3,4}      201: {2,19}
     15: {2,3}      119: {4,7}        203: {4,10}
     33: {2,5}      123: {2,13}       205: {3,13}
     35: {3,4}      135: {2,2,2,3}    207: {2,2,9}
     45: {2,2,3}    141: {2,15}       209: {5,8}
     51: {2,7}      143: {5,6}        215: {3,14}
     55: {3,5}      145: {3,10}       217: {4,11}
     69: {2,9}      153: {2,2,7}      219: {2,21}
     75: {2,3,3}    155: {3,11}       221: {6,7}
     77: {4,5}      161: {4,9}        225: {2,2,3,3}
     85: {3,7}      165: {2,3,5}      231: {2,4,5}
     91: {4,6}      175: {3,3,4}      245: {3,4,4}
     93: {2,11}     177: {2,17}       247: {6,8}
     95: {3,8}      187: {5,7}        249: {2,23}
     99: {2,2,5}    195: {2,3,6}      253: {5,9}

The complement is counted by A083710 (strict: A097986).
The complement with no 1's is A083711 (strict: A098965).
These partitions are counted by A338470 (strict: A341450).
The squarefree case is A339562, with squarefree complement A339563.
The case with maximum prime index not divisible by all others is A343338.
The case with maximum prime index divisible by all others is A343339.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A001221 counts distinct prime factors.
A006128 counts partitions with a selected position (strict: A015723).
A056239 adds up prime indices, row sums of A112798.
A299702 lists Heinz numbers of knapsack partitions.
A339564 counts factorizations with a selected factor.
Cf. A066637, A072774, A098743, A253249, A264401, A257993, A342050, A342051, A343344.

Select[Range[100],#==1||With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(p/Min@@p)]&]

A339562 Squarefree numbers with no prime index dividing all the other prime indices.

Original entry on oeis.org

1, 15, 33, 35, 51, 55, 69, 77, 85, 91, 93, 95, 105, 119, 123, 141, 143, 145, 155, 161, 165, 177, 187, 195, 201, 203, 205, 209, 215, 217, 219, 221, 231, 247, 249, 253, 255, 265, 285, 287, 291, 295, 299, 301, 309, 323, 327, 329, 335, 341, 345, 355, 357, 377, 381

Offset: 1

Gus Wiseman, Apr 10 2021

nonn

First differs from A342193 in lacking 45.
Alternative name: 1 and squarefree numbers with smallest prime index not dividing all the other prime indices.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Also 1 and Heinz numbers of strict integer partitions with smallest part not dividing all the others (counted by A341450). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      1: {}         141: {2,15}     219: {2,21}
     15: {2,3}      143: {5,6}      221: {6,7}
     33: {2,5}      145: {3,10}     231: {2,4,5}
     35: {3,4}      155: {3,11}     247: {6,8}
     51: {2,7}      161: {4,9}      249: {2,23}
     55: {3,5}      165: {2,3,5}    253: {5,9}
     69: {2,9}      177: {2,17}     255: {2,3,7}
     77: {4,5}      187: {5,7}      265: {3,16}
     85: {3,7}      195: {2,3,6}    285: {2,3,8}
     91: {4,6}      201: {2,19}     287: {4,13}
     93: {2,11}     203: {4,10}     291: {2,25}
     95: {3,8}      205: {3,13}     295: {3,17}
    105: {2,3,4}    209: {5,8}      299: {6,9}
    119: {4,7}      215: {3,14}     301: {4,14}
    123: {2,13}     217: {4,11}     309: {2,27}

The squarefree complement is A339563.
These partitions are counted by A341450.
The not necessarily squarefree version is A342193.
A000005 counts divisors.
A000070 counts partitions with a selected part.
A001221 counts distinct prime factors.
A005117 lists squarefree numbers.
A006128 counts partitions with a selected position (strict: A015723).
A056239 adds up prime indices (row sums of A112798).
A083710 counts partitions with a dividing part (strict: A097986).
Cf. A253249, A264401, A257993, A338470, A343337, A343338, A343339, A343379, A343380, A343382.

Select[Range[100],#==1||SquareFreeQ[#]&&With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(p/Min@@p)]&]

A083711 a(n) = A083710(n) - A000041(n-1).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 5, 3, 7, 1, 14, 1, 13, 8, 20, 1, 33, 1, 40, 14, 44, 1, 85, 6, 79, 25, 117, 1, 181, 1, 196, 45, 233, 17, 389, 1, 387, 80, 545, 1, 750, 1, 839, 165, 1004, 1, 1516, 12, 1612, 234, 2040, 1, 2766, 48, 3142, 388, 3720, 1, 5295, 1, 5606, 663, 7038, 83, 9194, 1, 10379, 1005

Offset: 1

N. J. A. Sloane, Jun 16 2003

Number of integer partitions of n with no 1's with a part dividing all the others. If n > 0, we can assume such a part is the smallest. - Gus Wiseman, Apr 18 2021

			From _Gus Wiseman_, Apr 18 2021: (Start)
The a(6) = 4 through a(12) = 13 partitions:
  (6)      (7)  (8)        (9)      (10)         (11)  (12)
  (3,3)         (4,4)      (6,3)    (5,5)              (6,6)
  (4,2)         (6,2)      (3,3,3)  (8,2)              (8,4)
  (2,2,2)       (4,2,2)             (4,4,2)            (9,3)
                (2,2,2,2)           (6,2,2)            (10,2)
                                    (4,2,2,2)          (4,4,4)
                                    (2,2,2,2,2)        (6,3,3)
                                                       (6,4,2)
                                                       (8,2,2)
                                                       (3,3,3,3)
                                                       (4,4,2,2)
                                                       (6,2,2,2)
                                                       (4,2,2,2,2)
                                                       (2,2,2,2,2,2)
(End)

L. M. Chawla, M. O. Levan and J. E. Maxfield, On a restricted partition function and its tables, J. Natur. Sci. and Math., 12 (1972), 95-101.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Allowing 1's gives A083710.
The strict case is A098965.
The complement (except also without 1's) is counted by A338470.
The dual version is A339619.
A000005 counts divisors.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
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. A001787, A001792, A015723, A097986, A098743, A130689, A130714, A264401, A339563, A342193.

with(combinat): with(numtheory): a := proc(n) c := 0: l := sort(convert(divisors(n), list)): for i from 1 to nops(l)-1 do c := c+numbpart(l[i]-1) od: RETURN(c): end: for j from 2 to 100 do printf(`%d,`,a(j)) od: # James Sellers, Jun 21 2003
# second Maple program:
a:= n-> max(1, add(combinat[numbpart](d-1), d=numtheory[divisors](n) minus {n})):
seq(a(n), n=1..69);  # Alois P. Heinz, Feb 15 2023

a[n_] := If[n==1, 1, Sum[PartitionsP[d-1], {d, Most@Divisors[n]}]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 15 2023 *)

a(n) = Sum_{ d|n, dA000041(d-1).

More terms from James Sellers, Jun 21 2003

A339742 Number of factorizations of n into distinct primes or squarefree semiprimes.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 4, 1, 0, 2, 2, 2, 1, 1, 2, 2, 0, 1, 4, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 3, 1, 2, 1, 0, 2, 4, 1, 1, 2, 4, 1, 0, 1, 2, 1, 1, 2, 4, 1, 0, 0, 2, 1, 3, 2, 2, 2, 0, 1, 3, 2, 1, 2, 2, 2, 0, 1, 1, 1, 1, 1, 4, 1, 0, 4

Offset: 1

Gus Wiseman, Dec 20 2020

nonn

A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct singletons or strict pairs, i.e., into a set of half-loops and edges;
(2) n can be factored into distinct primes or squarefree semiprimes.

			The a(n) factorizations for n = 6, 30, 60, 210, 420 are respectively 2, 4, 3, 10, 9:
  (6)    (5*6)    (6*10)    (6*35)     (2*6*35)
  (2*3)  (2*15)   (2*5*6)   (10*21)    (5*6*14)
         (3*10)   (2*3*10)  (14*15)    (6*7*10)
         (2*3*5)            (5*6*7)    (2*10*21)
                            (2*3*35)   (2*14*15)
                            (2*5*21)   (2*5*6*7)
                            (2*7*15)   (3*10*14)
                            (3*5*14)   (2*3*5*14)
                            (3*7*10)   (2*3*7*10)
                            (2*3*5*7)

Antti Karttunen, Table of n, a(n) for n = 1..69300
Index entries for sequences computed from exponents in factorization of n

Dirichlet convolution of A008966 with A339661.
A008966 allows only primes.
A339661 does not allow primes, only squarefree semiprimes.
A339740 lists the positions of zeros.
A339741 lists the positions of positive terms.
A339839 allows nonsquarefree semiprimes.
A339887 is the non-strict version.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A013929 cannot be factored into distinct primes.
A293511 are a product of distinct squarefree numbers in exactly one way.
A320663 counts non-isomorphic multiset partitions into singletons or pairs.
A339840 cannot be factored into distinct primes or semiprimes.
A339841 have exactly one factorization into primes or semiprimes.
The following count factorizations:
- A001055 into all positive integers > 1.
- A050320 into squarefree numbers.
- A050326 into distinct squarefree numbers.
- A320655 into semiprimes.
- A320656 into squarefree semiprimes.
- A320732 into primes or semiprimes.
- A322353 into distinct semiprimes.
- A339742 [this sequence] into distinct primes or squarefree semiprimes.
- A339839 into distinct primes or semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A000569 counts graphical partitions (A320922).
- A058696 counts all partitions of 2n (A300061).
- A209816 counts multigraphical partitions (A320924).
- A339656 counts loop-graphical partitions (A339658).
-
The following count partitions/factorizations of even length and give their Heinz numbers:
- A027187/A339846 has no additional conditions (A028260).
- A338914/A339562 can be partitioned into edges (A320911).
- A338916/A339563 can be partitioned into distinct pairs (A320912).
- A339559/A339564 cannot be partitioned into distinct edges (A320894).
- A339560/A339619 can be partitioned into distinct edges (A339561).
Cf. A000070, A001221, A005117, A320893, A320923, A338899, A339113, A339617, A353471.

sqps[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqps[n/d],Min@@#>d&]],{d,Select[Divisors[n],PrimeQ[#]||SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
Table[Length[sqps[n]],{n,100}]

A353471(n) = (numdiv(n)==2*omega(n));
A339742(n, u=(1+n)) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (dA353471(d), s += A339742(n/d, d))); (s)); \\ Antti Karttunen, May 02 2022

a(n) = Sum_{d|n squarefree} A339661(n/d).

More terms from Antti Karttunen, May 02 2022

cp's OEIS Frontend

A083710 Number of integer partitions of n with a part dividing all the other parts.

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A097986 Number of strict integer partitions of n with a part dividing all the other parts.

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A339561 Products of distinct squarefree semiprimes.

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

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

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A338916 Number of integer partitions of n that can be partitioned into distinct pairs of (possibly equal) parts.

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A342193 Numbers with no prime index dividing all the other prime indices.

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