A050320 -id:A050320 - cp's OEIS frontend

A376590 Second differences of consecutive squarefree numbers (A005117). First differences of A076259.

0, 1, -1, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 2, 0, -2, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 2, -2, 3, -2, 0, 0, -1, 0, 1, -1, 2, -2, 0, 1, -1, 0, 1, -1, 2, -2, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 1, 2, -3, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 2, -2, 2, -2, 3, -2, -1

Offset: 1

Gus Wiseman, Oct 01 2024

sign

			The squarefree numbers (A005117) are:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, ...
with first differences (A076259):
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, ...
with first differences (A376590):
  0, 1, -1, 0, 2, -2, 1, -1, 0, 1, 0, 0, -1, 0, 2, 0, -2, 0, 1, -1, 0, 1, -1, 0, 1, ...

The version for A000002 is A376604, first differences of A054354.
The first differences were A076259, see also A375927, A376305, A376306, A376307, A376311.
Zeros are A376591, complement A376592.
Sorted positions of first appearances are A376655.
A000040 lists the prime numbers, differences A001223.
A001597 lists perfect-powers, complement A007916.
A005117 lists squarefree numbers, complement A013929 (differences A078147).
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
A333254 lists run-lengths of differences between consecutive primes.
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376593 (nonsquarefree), A376596 (prime-power inclusive), A376599 (non-prime-power inclusive).
For squarefree numbers: A076259 (first differences), A376591 (inflections and undulations), A376592 (nonzero curvature), A376655 (sorted first positions).
Cf. A000961, A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A251092, A373198, A376342.

Differences[Select[Range[100],SquareFreeQ],2]

from math import isqrt
from sympy import mobius
def A376590(n):
    def iterfun(f,n=0):
        m, k = n, f(n)
        while m != k: m, k = k, f(k)
        return m
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    a = iterfun(f,n)
    b = iterfun(lambda x:f(x)+1,a)
    return a+iterfun(lambda x:f(x)+2,b)-(b<<1) # Chai Wah Wu, Oct 02 2024

A381806 Numbers that cannot be written as a product of squarefree numbers with distinct sums of prime indices.

Original entry on oeis.org

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336

Offset: 1

Gus Wiseman, Mar 12 2025

nonn

First differs from A212164 in having 3600.
First differs from A293243 in having 18000.
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 numbers whose prime indices cannot be partitioned into a multiset of sets with distinct sums.

			There are 4 factorizations of 18000 into squarefree numbers:
  (2*2*3*5*10*30)
  (2*2*5*6*10*15)
  (2*2*10*15*30)
  (2*5*6*10*30)
but none of these has all distinct sums of prime indices, so 18000 is in the sequence.

Strongly normal multisets of this type are counted by A292444.
These are the zeros in A381633, see A050320, A321469, A381078, A381634.
For distinct blocks see A050326, A293243, A293511, A358914, A381441.
For more on set multipartitions see A089259, A116540, A270995, A296119, A318360.
For more on set multipartitions with distinct sums see A279785, A381718.
For constant instead of strict blocks we have A381636, see A381635, A381716.
Partitions of this type are counted by A381990, complement A381992.
The complement is A382075.
A001055 counts multiset partitions, strict A045778.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000688, A000720, A001222, A005117, A299202, A300385, A381454, A381870.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
sqfics[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfics[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]]
Select[Range[nn],Length[Select[sqfics[#],UnsameQ@@hwt/@#&]]==0&]

A377046 Array read by downward antidiagonals where A(n,k) is the n-th term of the k-th differences of nonsquarefree numbers.

Original entry on oeis.org

4, 8, 4, 9, 1, -3, 12, 3, 2, 5, 16, 4, 1, -1, -6, 18, 2, -2, -3, -2, 4, 20, 2, 0, 2, 5, 7, 3, 24, 4, 2, 2, 0, -5, -12, -15, 25, 1, -3, -5, -7, -7, -2, 10, 25, 27, 2, 1, 4, 9, 16, 23, 25, 15, -10, 28, 1, -1, -2, -6, -15, -31, -54, -79, -94, -84, 32, 4, 3, 4, 6, 12, 27, 58, 112, 191, 285, 369

Offset: 0

Gus Wiseman, Oct 19 2024

Row k is the k-th differences of A013929.

			Array form:
        n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:  n=9:
  ---------------------------------------------------------
  k=0:   4     8     9    12    16    18    20    24    25
  k=1:   4     1     3     4     2     2     4     1     2
  k=2:  -3     2     1    -2     0     2    -3     1    -1
  k=3:   5    -1    -3     2     2    -5     4    -2     4
  k=4:  -6    -2     5     0    -7     9    -6     6    -7
  k=5:   4     7    -5    -7    16   -15    12   -13    10
  k=6:   3   -12    -2    23   -31    27   -25    23   -13
  k=7: -15    10    25   -54    58   -52    48   -36    13
  k=8:  25    15   -79   112  -110   100   -84    49     1
  k=9: -10   -94   191  -222   210  -184   133   -48   -57
Triangle form:
   4
   8   4
   9   1  -3
  12   3   2   5
  16   4   1  -1  -6
  18   2  -2  -3  -2   4
  20   2   0   2   5   7   3
  24   4   2   2   0  -5 -12 -15
  25   1  -3  -5  -7  -7  -2  10  25
  27   2   1   4   9  16  23  25  15 -10
  28   1  -1  -2  -6 -15 -31 -54 -79 -94 -84
  32   4   3   4   6  12  27  58 112 191 285 369

Initial rows: A013929, A078147, A376593.
The version for primes is A095195, noncomposites A376682, composites A377033.
A version for partitions is A175804, cf. A053445, A281425, A320590.
For squarefree numbers we have A377038, sums A377039, absolute A377040.
Triangle row-sums are A377047, absolute version A377048.
Column n = 1 is A377049, for squarefree A377041, for prime A007442 or A030016.
First position of 0 in each row is A377050.
For prime-power instead of nonsquarefree we have A377051.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers.
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
Cf. A000961, A007674, A053797, A053806, A061398, A072284, A112925, A120992, A376311, A376591, A376592.

nn=9;
t=Table[Take[Differences[NestList[NestWhile[#+1&,#+1,SquareFreeQ[#]&]&,4,2*nn],k],nn],{k,0,nn}]
Table[t[[j,i-j+1]],{i,nn},{j,i}]

A(i,j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) A013929(i+k).

A381992 Number of integer partitions of n that can be partitioned into sets with distinct sums.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 9, 13, 17, 25, 33, 44, 59, 77, 100, 134, 170, 217, 282, 360, 449, 571, 719, 899, 1122, 1391, 1727, 2136, 2616, 3209, 3947, 4800, 5845, 7094, 8602, 10408, 12533, 15062, 18107, 21686, 25956, 30967, 36936, 43897, 52132, 61850, 73157, 86466, 101992, 120195

Offset: 0

Gus Wiseman, Mar 16 2025

nonn

Also the number of integer partitions of n whose Heinz number belongs to A382075 (can be written as a product of squarefree numbers with distinct sums of prime indices).

			There are 6 ways to partition (3,2,2,1) into sets:
  {{2},{1,2,3}}
  {{1,2},{2,3}}
  {{1},{2},{2,3}}
  {{2},{2},{1,3}}
  {{2},{3},{1,2}}
  {{1},{2},{2},{3}}
Of these, 3 have distinct block sums:
  {{2},{1,2,3}}
  {{1,2},{2,3}}
  {{1},{2},{2,3}}
so (3,2,2,1) is counted under a(8).
The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)  (3)    (4)      (5)      (6)        (7)        (8)
            (2,1)  (3,1)    (3,2)    (4,2)      (4,3)      (5,3)
                   (2,1,1)  (4,1)    (5,1)      (5,2)      (6,2)
                            (2,2,1)  (3,2,1)    (6,1)      (7,1)
                            (3,1,1)  (4,1,1)    (3,2,2)    (3,3,2)
                                     (2,2,1,1)  (3,3,1)    (4,2,2)
                                                (4,2,1)    (4,3,1)
                                                (5,1,1)    (5,2,1)
                                                (3,2,1,1)  (6,1,1)
                                                           (3,2,2,1)
                                                           (3,3,1,1)
                                                           (4,2,1,1)
                                                           (3,2,1,1,1)

More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
Twice-partitions of this type are counted by A279785.
Multiset partitions of this type are counted by A381633, zeros of A381634.
For constant instead of strict blocks see A381717, A381636, A381635, A381716, A381991.
Normal multiset partitions of this type are counted by A381718, see A116539.
The complement is counted by A381990, ranked by A381806.
These partitions are ranked by A382075.
For distinct blocks instead of sums we have A382077, complement A382078.
For a unique choice we have A382079.
A000041 counts integer partitions, strict A000009.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions of prime indices into distinct sets.
A265947 counts refinement-ordered pairs of integer partitions.
A382201 lists MM-numbers of sets with distinct sums.
Cf. A002846, A047966, A213427, A279786, A299202, A317142, A381870.
Cf. A293243, A293511, A358914, A381078, A381441, A381454.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[IntegerPartitions[n],Length[Select[mps[#], And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]>0&]],{n,0,10}]

a(21)-a(50) from Bert Dobbelaere, Mar 29 2025

A174725 a(n) = (A074206(n) + A008683(n))/2.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 4, 0, 2, 2, 4, 0, 4, 0, 4, 2, 2, 0, 10, 1, 2, 2, 4, 0, 6, 0, 8, 2, 2, 2, 13, 0, 2, 2, 10, 0, 6, 0, 4, 4, 2, 0, 24, 1, 4, 2, 4, 0, 10, 2, 10, 2, 2, 0, 22, 0, 2, 4, 16, 2, 6, 0, 4, 2, 6, 0, 38, 0, 2, 4, 4, 2

Offset: 1

Mats Granvik, Mar 28 2010

nonn

From Mats Granvik, May 25 2017: (Start)
A074206(n) = A002033(n-1) = a(n) + A174726(n).
A008683(n) = a(n) - A174726(n).
Let m = size of matrix a matrix T, and let T be defined as follows:
T(n,k) = if m = 1 then 1 else if mod(n, k) = 0 then if and(n = k, n = m) then 0 else 1 else if and(n = 1, k = m) then 1 else 0
a(n) is then the number of permutation matrices with a positive contribution in the determinant of matrix T. The determinant of T is equal to the Möbius function A008683, see Mathematica program below for how to compute the determinant.
A174726 is the number of permutation matrices with a negative contribution in the determinant of matrix T.
(End)
From Gus Wiseman, Jan 04 2021: (Start)
Also the number of ordered factorizations of n into an even number of factors > 1. The non-ordered case is A339846. For example, the a(n) factorizations for n = 12, 24, 30, 32, 36 are:
  (2*6)  (3*8)      (5*6)   (4*8)      (4*9)
  (3*4)  (4*6)      (6*5)   (8*4)      (6*6)
  (4*3)  (6*4)      (10*3)  (16*2)     (9*4)
  (6*2)  (8*3)      (15*2)  (2*16)     (12*3)
         (12*2)     (2*15)  (2*2*2*4)  (18*2)
         (2*12)     (3*10)  (2*2*4*2)  (2*18)
         (2*2*2*3)          (2*4*2*2)  (3*12)
         (2*2*3*2)          (4*2*2*2)  (2*2*3*3)
         (2*3*2*2)                     (2*3*2*3)
         (3*2*2*2)                     (2*3*3*2)
                                       (3*2*2*3)
                                       (3*2*3*2)
                                       (3*3*2*2)
(End)

Mats Granvik, Table of n, a(n) for n = 1..10000

Cf. A008683, A051731.
The odd version is A174726.
The unordered version is A339846.
A001055 counts factorizations, with strict case A045778.
A058696 counts partitions of even numbers, ranked by A300061.
A074206 counts ordered factorizations, with strict case A254578.
A251683 counts ordered factorizations by product and length.
Other cases of even length:
- A024430 counts set partitions of even length.
- A027187 counts partitions of even length.
- A034008 counts compositions of even length.
- A052841 counts ordered set partitions of even length.
- A067661 counts strict partitions of even length.
- A332305 counts strict compositions of even length
Cf. A002033, A027193, A028260, A050320, A058695, A236913, A316439, A320655, A320656, A339890, A378216 (Dirichlet inverse).

(* From Mats Granvik, May 25 2017: (Start) *)
Clear[t, nn]; nn = 77; t[1, 1] = 1; t[n_, k_] := t[n, k] = If[k == 1, Sum[t[n, k + i], {i, 1, n - 1}], If[Mod[n, k] == 0, t[n/k, 1], 0], 0]; Monitor[Table[Sum[If[Mod[n, k] == 0, MoebiusMu[k]*t[n/k, 1], 0], {k, 1, 77}], {n, 1, nn}], n]
(* The Möbius function as a determinant *) Table[Det[Table[Table[If[m == 1, 1, If[Mod[n, k] == 0, If[And[n == k, n == m], 0, 1], If[And[n == 1, k == m], 1, 0]]], {k, 1, m}], {n, 1, m}]], {m, 1, 42}]
(* (End) *)
ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
Table[Length[Select[ordfacs[n],EvenQ@*Length]],{n,100}] (* Gus Wiseman, Jan 04 2021 *)

a(n) = (Mobius transform of a(n)) + (Mobius transform of A174726). - Mats Granvik, Apr 04 2010
From Mats Granvik, May 25 2017: (Start)
This sequence is the Moebius transform of A074206.
a(n) = (A074206(n) + A008683(n))/2.
(End)
G.f. A(x) satisfies: A(x) = x + Sum_{i>=2} Sum_{j>=2} A(x^(i*j)). - Ilya Gutkovskiy, May 11 2019

References to A002033(n-1) changed to A074206(n) by Antti Karttunen, Nov 23 2024

A339741 Products of distinct primes or squarefree semiprimes.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84

Offset: 1

Gus Wiseman, Dec 23 2020

nonn

First differs from A212167 in lacking 1080, with prime indices {1,1,1,2,2,2,3}.
First differs from A335433 in lacking 72 (see example).
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 and strict pairs, i.e., into a set of half-loops and edges;
(2) n can be factored into distinct primes or squarefree semiprimes;
(3) the prime signature of n is half-loop-graphical.

			The sequence of terms together with their prime indices begins:
       1: {}           20: {1,1,3}        39: {2,6}
       2: {1}          21: {2,4}          41: {13}
       3: {2}          22: {1,5}          42: {1,2,4}
       5: {3}          23: {9}            43: {14}
       6: {1,2}        26: {1,6}          44: {1,1,5}
       7: {4}          28: {1,1,4}        45: {2,2,3}
      10: {1,3}        29: {10}           46: {1,9}
      11: {5}          30: {1,2,3}        47: {15}
      12: {1,1,2}      31: {11}           50: {1,3,3}
      13: {6}          33: {2,5}          51: {2,7}
      14: {1,4}        34: {1,7}          52: {1,1,6}
      15: {2,3}        35: {3,4}          53: {16}
      17: {7}          36: {1,1,2,2}      55: {3,5}
      18: {1,2,2}      37: {12}           57: {2,8}
      19: {8}          38: {1,8}          58: {1,10}
For example, we have 36 = (2*3*6), so 36 is in the sequence. On the other hand, a complete list of all strict factorizations of 72 is: (2*3*12), (2*4*9), (2*36), (3*4*6), (3*24), (4*18), (6*12), (8*9), (72). Since none of these consists of only primes or squarefree semiprimes, 72 is not in the sequence. A complete list of all factorizations of 1080 into primes or squarefree semiprimes is:
  (2*2*2*3*3*3*5)
  (2*2*2*3*3*15)
  (2*2*3*3*3*10)
  (2*2*3*3*5*6)
  (2*2*3*6*15)
  (2*3*3*6*10)
  (2*3*5*6*6)
  (2*6*6*15)
  (3*6*6*10)
  (5*6*6*6)
Since none of these is strict, 1080 is not in the sequence.

Eric Weisstein's World of Mathematics, Degree Sequence.
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

See link for additional cross-references.
Allowing only primes gives A013929.
Not allowing primes gives A339561.
Complement of A339740.
Positions of positive terms in A339742.
Allowing squares of primes gives the complement of A339840.
Unlabeled multiset partitions of this type are counted by A339888.
A001055 counts factorizations.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A339841 have exactly one factorization into primes or semiprimes.
Cf. A001221, A005117, A028260, A030229, A050320, A112798, A309356, A320663, A320893, A320924, A338899.

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

A381078 Number of multisets that can be obtained by partitioning the prime indices of n into a multiset of sets (set multipartition) and taking their sums.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 5, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 6, 1, 2, 2, 1, 2, 5, 1, 2, 2, 5, 1, 3, 1, 2, 2, 2, 2, 5, 1, 2, 1, 2, 1, 6, 2, 2, 2

Offset: 1

Gus Wiseman, Mar 05 2025

nonn

First differs from A050320 at a(210) = 13, A050320(210) = 15. This comes from the set multipartitions {{3},{1,2,4}} and {{1,2},{3,4}}, and from {{4},{1,2,3}} and {{1,3},{2,4}}.
Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a factorization of n into squarefree numbers > 1.
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.
A multiset partition can be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
Set multipartitions are generally not transitive. For example, we have arrows: {{1},{1,2}}: {1,1,2} -> {1,3} and {{1,3}}: {1,3} -> {4}, but there is no set multipartition {1,1,2} -> {4}.

			The prime indices of 60 are {1,1,2,3}, with set multipartitions:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{1},{3},{1,2}}
  {{1},{1},{2},{3}}
with block-sums: {1,6}, {3,4}, {1,1,5}, {1,2,4}, {1,3,3}, {1,1,2,3}, which are all different multisets, so a(60) = 6.

Robert Price, Table of n, a(n) for n = 1..1000

Before taking sums we had A050320, strict A050326 (zeros A293243), distinct sums A381633.
For distinct blocks we have A381441.
The lower version is A381454.
For distinct block-sums we have A381634.
Other multiset partitions of prime indices:
- For multisets of constant multisets (A000688) see A381455 (upper), A381453 (lower).
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For strict multiset partitions (A045778) see A381452.
- For sets of constant multisets (A050361) see A381717.
- For strict multiset partitions with distinct sums (A321469) see A381637.
- For sets of constant multisets with distinct sums (A381635) see A381716, A381636.
More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A002846, A005117, A025487, A066328, A213242, A213385, A213427, A299201, A299202, A300385.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[sqfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[Union[Sort[hwt/@#]&/@sqfacs[n]]],{n,100}]

a(A002110(n)) = A066723(n).

A259936 Number of ways to express the integer n as a product of its unitary divisors (A034444).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 5, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 5, 1, 2, 2, 1, 2, 5, 1, 2, 2, 5, 1, 2, 1, 2, 2, 2, 2, 5, 1, 2, 1, 2, 1, 5, 2, 2, 2, 2, 1, 5, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 5, 1, 2, 5

Offset: 1

Geoffrey Critzer, Jul 09 2015

nonn

Equivalently, a(n) is the number of ways to express the cyclic group Z_n as a direct sum of its Hall subgroups.  A Hall subgroup of a finite group G is a subgroup whose order is coprime to its index.
a(n) is the number of ways to partition the set of distinct prime factors of n.
Also the number of singleton or pairwise coprime factorizations of n. - Gus Wiseman, Sep 24 2019

			a(60) = 5 because we have: 60 = 4*3*5 = 4*15 = 3*20 = 5*12.
For n = 36, its unitary divisors are 1, 4, 9, 36. From these we obtain 36 either as 1*36 or 4*9, thus a(36) = 2. - _Antti Karttunen_, Oct 21 2017

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Wikipedia, Hall subgroup
Index entries for sequences computed from exponents in factorization of n

Cf. A000110, A001055, A001221, A034444, A089233, A258466, A281116, A285572.
Differs from A050320 for the first time at n=36.
Differs from A354870 for the first time at n=210, where a(210) = 15, while A354870(210) = 12.
Cf. A304716, A302569, A304711, A305079.
Related classes of factorizations:
- No conditions: A001055
- Strict: A045778
- Constant: A089723
- Distinct multiplicities: A255231
- Singleton or coprime: A259936
- Relatively prime: A281116
- Aperiodic: A303386
- Stable (indivisible): A305149
- Connected: A305193
- Strict relatively prime: A318721
- Uniform: A319269
- Intersecting: A319786
- Constant or distinct factors coprime: A327399
- Constant or relatively prime: A327400
- Coprime: A327517
- Not relatively prime: A327658
- Distinct factors coprime: A327695

map(combinat:-bell @ nops @ numtheory:-factorset, [$1..100]); # Robert Israel, Jul 09 2015

Table[BellB[PrimeNu[n]], {n, 1, 75}]
(* second program *)
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Length[#]==1||CoprimeQ@@#&]],{n,100}] (* Gus Wiseman, Sep 24 2019 *)

a(n) = my(t=omega(n), x='x, m=contfracpnqn(matrix(2, t\2, y, z, if( y==1, -z*x^2, 1 - (z+1)*x)))); polcoeff(1/(1 - x + m[2, 1]/m[1, 1]) + O(x^(t+1)), t) \\ Charles R Greathouse IV, Jun 30 2017

a(n) = A000110(A001221(n)).

a(n > 1) = A327517(n) + 1. - Gus Wiseman, Sep 24 2019

Incorrect comment removed by Antti Karttunen, Jun 11 2022

A377038 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the squarefree numbers.

Original entry on oeis.org

1, 2, 1, 3, 1, 0, 5, 2, 1, 1, 6, 1, -1, -2, -3, 7, 1, 0, 1, 3, 6, 10, 3, 2, 2, 1, -2, -8, 11, 1, -2, -4, -6, -7, -5, 3, 13, 2, 1, 3, 7, 13, 20, 25, 22, 14, 1, -1, -2, -5, -12, -25, -45, -70, -92, 15, 1, 0, 1, 3, 8, 20, 45, 90, 160, 252, 17, 2, 1, 1, 0, -3, -11, -31, -76, -166, -326, -578

Offset: 0

Gus Wiseman, Oct 18 2024

Row n is the k-th differences of A005117 = the squarefree numbers.

			Array form:
        n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:  n=9:
  ----------------------------------------------------------
  k=0:   1     2     3     5     6     7    10    11    13
  k=1:   1     1     2     1     1     3     1     2     1
  k=2:   0     1    -1     0     2    -2     1    -1     0
  k=3:   1    -2     1     2    -4     3    -2     1     1
  k=4:  -3     3     1    -6     7    -5     3     0    -2
  k=5:   6    -2    -7    13   -12     8    -3    -2     3
  k=6:  -8    -5    20   -25    20   -11     1     5    -5
  k=7:   3    25   -45    45   -31    12     4   -10    10
  k=8:  22   -70    90   -76    43    -8   -14    20   -19
  k=9: -92   160  -166   119   -51    -6    34   -39    28
Triangle form:
   1
   2   1
   3   1   0
   5   2   1   1
   6   1  -1  -2  -3
   7   1   0   1   3   6
  10   3   2   2   1  -2  -8
  11   1  -2  -4  -6  -7  -5   3
  13   2   1   3   7  13  20  25  22
  14   1  -1  -2  -5 -12 -25 -45 -70 -92
  15   1   0   1   3   8  20  45  90 160 252

Row k=0 is A005117.
Row k=1 is A076259.
Row k=2 is A376590.
The version for primes is A095195, noncomposites A376682, composites A377033.
A version for partitions is A175804, cf. A053445, A281425, A320590.
Triangle row-sums are A377039, absolute version A377040.
Column n = 1 is A377041, for primes A007442 or A030016.
First position of 0 in each row is A377042.
For nonsquarefree instead of squarefree numbers we have A377046.
For prime-powers instead of squarefree numbers we have A377051.
A000040 lists the primes, differences A001223, seconds A036263.
A005117 lists the squarefree numbers, complement A013929 (differences A078147).
A073576 counts integer partitions into squarefree numbers, factorizations A050320.
Cf. A007674, A053797, A053806, A061398, A072284, A075526, A112925, A112926, A120992, A373198, A376311, A376591, A376592.

nn=9;
t=Table[Take[Differences[NestList[NestWhile[#+1&,#+1,!SquareFreeQ[#]&]&,1,2*nn],k],nn],{k,0,nn}]
Table[t[[j,i-j+1]],{i,nn},{j,i}]

A(i,j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) A005117(i+k).

A381453 Number of multisets that can be obtained by choosing a constant integer partition of each prime index of n and taking the multiset union.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 08 2025

nonn

First differs from A355733 and A355735 at a(21) = 6, A355733(21) = A355735(21) = 5.
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.
A multiset partition can be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
Multisets of constant multisets are generally not transitive. For example, we have arrows: {{1,1},{2}}: {1,1,2} -> {2,2} and {{2,2}}: {2,2} -> {4}, but there is no multiset of constant multisets {1,1,2} -> {4}.

			The a(21) = 6 multisets are: {2,4}, {1,1,4}, {2,2,2}, {1,1,2,2}, {2,1,1,1,1}, {1,1,1,1,1,1}.
The a(n) partitions for n = 1, 3, 7, 13, 53, 21 (G = 16):
  ()  (2)   (4)     (6)       (G)                 (42)
      (11)  (22)    (33)      (88)                (411)
            (1111)  (222)     (4444)              (222)
                    (111111)  (22222222)          (2211)
                              (1111111111111111)  (21111)
                                                  (111111)

Robert Price, Table of n, a(n) for n = 1..300

Positions of 1 are A000079.
The strict case is A008966.
Before sorting we had A355731.
Choosing divisors instead of constant multisets gives A355733.
The upper version is A381455, before taking sums A000688.
Multiset partitions of prime indices:
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For strict multiset partitions (A045778) see A381452.
- For set multipartitions (A050320) see A381078 (upper), A381454 (lower).
- For set systems (A050326, zeros A293243) see A381441 (upper).
- For sets of constant multisets (A050361) see A381715.
- For strict multiset partitions with distinct sums (A321469) see A381637.
- For set systems with distinct sums (A381633, zeros A381806) see A381634.
- For sets of constant multisets with distinct sums (A381635, zeros A381636) see A381716.
More on multiset partitions into constant blocks: A006171, A279784, A295935.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A000961, A001222, A002577, A018818, A213242, A213385, A213427, A275870, A299200, A300273, A300385.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@prix[n]]]],{n,nn}]

a(A002110(n)) = A381807(n).

cp's OEIS Frontend

A376590 Second differences of consecutive squarefree numbers (A005117). First differences of A076259.

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A381806 Numbers that cannot be written as a product of squarefree numbers with distinct sums of prime indices.

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A377046 Array read by downward antidiagonals where A(n,k) is the n-th term of the k-th differences of nonsquarefree numbers.

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A381992 Number of integer partitions of n that can be partitioned into sets with distinct sums.

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A174725 a(n) = (A074206(n) + A008683(n))/2.

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A339741 Products of distinct primes or squarefree semiprimes.

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A381078 Number of multisets that can be obtained by partitioning the prime indices of n into a multiset of sets (set multipartition) and taking their sums.

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A259936 Number of ways to express the integer n as a product of its unitary divisors (A034444).

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