A143773 -id:A143773 - cp's OEIS frontend

A319333 Heinz numbers of integer partitions whose sum is equal to their LCM.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 198, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263

Offset: 1

Gus Wiseman, Sep 17 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of partitions whose Heinz numbers are in the sequence begins: (1), (2), (3), (4), (5), (6), (7), (8), (9), (10), (3,2,1), (11), (12).

Cf. A067538, A074761, A143773, A290103, A305566, A316429, A316431, A316432, A316433, A317624, A319315, A319334.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],LCM@@primeMS[#]==Total[primeMS[#]]&]

A340851 Number of factorizations of n such that every factor is a divisor of the number of factors.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

Also factorizations whose number of factors is divisible by their least common multiple.

			The a(n) factorizations for n = 8192, 46656, 73728:
  2*2*2*2*2*4*8*8          6*6*6*6*6*6              2*2*2*2*2*2*2*2*2*4*6*6
  2*2*2*2*4*4*4*8          2*2*2*2*2*2*3*3*3*3*3*3  2*2*2*2*2*2*2*2*3*4*4*6
  2*2*2*4*4*4*4*4                                   2*2*2*2*2*2*2*3*3*4*4*4
  2*2*2*2*2*2*2*2*2*2*2*4                           2*2*2*2*2*2*2*2*2*2*6*12
                                                    2*2*2*2*2*2*2*2*2*3*4*12

The version for partitions is A340693, with reciprocal version A143773.
Positions of nonzero terms are A340852.
The reciprocal version is A340853.
A320911 can be factored into squarefree semiprimes.
A340597 have an alt-balanced factorization.
A340656 lack a twice-balanced factorization, complement A340657.
- Factorizations -
A001055 counts factorizations, with strict case A045778.
A316439 counts factorizations by product and length.
A339846 counts factorizations of even length.
A339890 counts factorizations of odd length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340653 counts balanced factorizations.
A340785 counts factorizations into even numbers, even-length case A340786.
A340831/A340832 count factorizations with odd maximum/minimum.
A340854 cannot be factored with odd least factor, complement A340855.
Cf. A067538, A074761, A168659, A301987, A327517, A340596, A340599, A340654, A340655, A340827, A340830.

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],And@@IntegerQ/@(Length[#]/#)&]],{n,100}]

A340853 Number of factorizations of n such that every factor is a multiple of the number of factors.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

Also factorizations whose greatest common divisor is a multiple of the number of factors.

			The a(n) factorizations for n = 2, 4, 16, 48, 96, 144, 216, 240, 432:
  2   4     16    48     96     144     216      240     432
      2*2   2*8   6*8    2*48   2*72    4*54     4*60    6*72
            4*4   2*24   4*24   4*36    6*36     6*40    8*54
                  4*12   6*16   6*24    12*18    8*30    12*36
                         8*12   8*18    2*108    10*24   18*24
                                12*12   6*6*6    12*20   2*216
                                        3*3*24   2*120   4*108
                                        3*6*12           3*3*48
                                                         3*6*24
                                                         6*6*12
                                                         3*12*12

Positions of 1's are A048103.
Positions of terms > 1 are A100716.
The version for partitions is A143773 (A316428).
The reciprocal for partitions is A340693 (A340606).
The version for strict partitions is A340830.
The reciprocal version is A340851.
A320911 can be factored into squarefree semiprimes.
A340597 have an alt-balanced factorization.
A340656 lack a twice-balanced factorization, complement A340657.
- Factorizations -
A001055 counts factorizations, with strict case A045778.
A316439 counts factorizations by product and length.
A339846 counts factorizations of even length.
A339890 counts factorizations of odd length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340653 counts balanced factorizations.
A340785 counts factorizations into even factors, even-length case A340786.
A340831/A340832 counts factorizations with odd maximum/minimum.
A340854 cannot be factored with odd least factor, complement A340855.
Cf. A067538, A168659, A301987, A316413, A327517, A340596, A340599, A340654, A340655, A340827.

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],n>1&&Divisible[GCD@@#,Length[#]]&]],{n,100}]

A319567 Product of y divided by the GCD of y to the power of the length of y, where y is the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 4, 6, 1, 1, 4, 1, 3, 2, 5, 1, 2, 1, 6, 1, 4, 1, 6, 1, 1, 10, 7, 12, 4, 1, 8, 3, 3, 1, 8, 1, 5, 12, 9, 1, 2, 1, 9, 14, 6, 1, 8, 15, 4, 4, 10, 1, 6, 1, 11, 2, 1, 2, 10, 1, 7, 18, 12, 1, 4, 1, 12, 18, 8, 20, 12, 1, 3, 1, 13

Offset: 0

Gus Wiseman, Sep 23 2018

nonn

The Heinz number of a partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Cf. A003963, A001222, A056239, A067538, A112798, A143773, A289508, A289509, A316430.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,1,Times@@primeMS[n]/GCD@@primeMS[n]^PrimeOmega[n]],{n,100}]

a(n) = A003963(n) / A289508(n) ^ A001222(n).

A340693 Number of integer partitions of n where each part is a divisor of the number of parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 2, 5, 5, 7, 7, 10, 10, 14, 14, 17, 19, 24, 24, 32, 33, 42, 43, 58, 59, 75, 79, 98, 104, 124, 128, 156, 166, 196, 204, 239, 251, 292, 306, 352, 372, 426, 445, 514, 543, 616, 652, 745, 790, 896, 960, 1080, 1162, 1311, 1400, 1574, 1692, 1892

Offset: 0

Gus Wiseman, Jan 23 2021

nonn

The only strict partitions counted are (), (1), and (2,1).

Is there a simple generating function?

			The a(1) = 1 through a(9) = 7 partitions:
  1  11  21   22    311    2211    331      2222      333
         111  1111  2111   111111  2221     4211      4221
                    11111          4111     221111    51111
                                   211111   311111    222111
                                   1111111  11111111  321111
                                                      21111111
                                                      111111111

Note: Heinz numbers are given in parentheses below.
The reciprocal version is A143773 (A316428), with strict case A340830.
The case where length also divides n is A326842 (A326847).
The Heinz numbers of these partitions are A340606.
The version for factorizations is A340851, with reciprocal version A340853.
A018818 counts partitions of n into divisors of n (A326841).
A047993 counts balanced partitions (A106529).
A067538 counts partitions of n whose length/max divides n (A316413/A326836).
A067539 counts partitions with integer geometric mean (A326623).
A072233 counts partitions by sum and length.
A168659 = partitions whose greatest part divides their length (A340609).
A168659 = partitions whose length divides their greatest part (A340610).
A326843 = partitions of n whose length and maximum both divide n (A326837).
A330950 = partitions of n whose Heinz number is divisible by n (A324851).
Cf. A000041, A003114, A006141, A033630, A064174, A074761, A102627, A200750, A237984, A298423, A340827.

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

A343662 Irregular triangle read by rows where T(n,k) is the number of strict length k chains of divisors of n, 0 <= k <= Omega(n) + 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 4, 5, 2, 1, 2, 1, 1, 4, 6, 4, 1, 1, 3, 3, 1, 1, 4, 5, 2, 1, 2, 1, 1, 6, 12, 10, 3, 1, 2, 1, 1, 4, 5, 2, 1, 4, 5, 2, 1, 5, 10, 10, 5, 1, 1, 2, 1, 1, 6, 12, 10, 3, 1, 2, 1, 1, 6, 12, 10, 3, 1, 4, 5, 2, 1, 4, 5, 2

Offset: 1

Gus Wiseman, May 01 2021

			Triangle begins:
   1:  1  1
   2:  1  2  1
   3:  1  2  1
   4:  1  3  3  1
   5:  1  2  1
   6:  1  4  5  2
   7:  1  2  1
   8:  1  4  6  4  1
   9:  1  3  3  1
  10:  1  4  5  2
  11:  1  2  1
  12:  1  6 12 10  3
  13:  1  2  1
  14:  1  4  5  2
  15:  1  4  5  2
  16:  1  5 10 10  5  1
For example, row n = 12 counts the following chains:
  ()  (1)   (2/1)   (4/2/1)   (12/4/2/1)
      (2)   (3/1)   (6/2/1)   (12/6/2/1)
      (3)   (4/1)   (6/3/1)   (12/6/3/1)
      (4)   (4/2)   (12/2/1)
      (6)   (6/1)   (12/3/1)
      (12)  (6/2)   (12/4/1)
            (6/3)   (12/4/2)
            (12/1)  (12/6/1)
            (12/2)  (12/6/2)
            (12/3)  (12/6/3)
            (12/4)
            (12/6)

Column k = 1 is A000005.
Row ends are A008480.
Row lengths are A073093.
Column k = 2 is A238952.
The case from n to 1 is A334996 or A251683 (row sums: A074206).
A non-strict version is A334997 (transpose: A077592).
The case starting with n is A337255 (row sums: A067824).
Row sums are A337256 (nonempty: A253249).
A001055 counts factorizations.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A097805 counts compositions by sum and length.
A122651 counts strict chains of divisors summing to n.
A146291 counts divisors of n with k prime factors (with multiplicity).
A163767 counts length n - 1 chains of divisors of n.
A167865 counts strict chains of divisors > 1 summing to n.
A337070 counts strict chains of divisors starting with superprimorials.
Cf. A002033, A007425, A007426, A051026, A062319, A143773, A186972, A327527, A337074, A337105, A337107, A343658.

Table[Length[Select[Reverse/@Subsets[Divisors[n],{k}],And@@Divisible@@@Partition[#,2,1]&]],{n,15},{k,0,PrimeOmega[n]+1}]

A353395 Numbers k such that the prime shadow of k equals the product of prime shadows of the prime indices of k.

Original entry on oeis.org

1, 3, 5, 11, 15, 17, 26, 31, 33, 41, 51, 55, 58, 59, 67, 78, 83, 85, 86, 93, 94, 109, 123, 126, 127, 130, 146, 148, 155, 157, 158, 165, 174, 177, 179, 187, 191, 196, 201, 202, 205, 211, 241, 244, 249, 255, 258, 274, 277, 278, 282, 283, 284, 286, 290, 295, 298

Offset: 1

Gus Wiseman, May 17 2022

nonn

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.

We define the prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.

			The terms together with their prime indices begin:
      1: {}         78: {1,2,6}      158: {1,22}
      3: {2}        83: {23}         165: {2,3,5}
      5: {3}        85: {3,7}        174: {1,2,10}
     11: {5}        86: {1,14}       177: {2,17}
     15: {2,3}      93: {2,11}       179: {41}
     17: {7}        94: {1,15}       187: {5,7}
     26: {1,6}     109: {29}         191: {43}
     31: {11}      123: {2,13}       196: {1,1,4,4}
     33: {2,5}     126: {1,2,2,4}    201: {2,19}
     41: {13}      127: {31}         202: {1,26}
     51: {2,7}     130: {1,3,6}      205: {3,13}
     55: {3,5}     146: {1,21}       211: {47}
     58: {1,10}    148: {1,1,12}     241: {53}
     59: {17}      155: {3,11}       244: {1,1,18}
     67: {19}      157: {37}         249: {2,23}
For example, 126 is in the sequence because its prime indices {1,2,2,4} have shadows {1,2,2,3}, with product 12, which is also the prime shadow of 126.

The prime terms are A006450.
The LHS (prime shadow) is A181819, with an inverse A181821.
The RHS (product of shadows) is A353394, first appearances A353397.
This is a ranking of the partitions counted by A353396.
Another related comparison is A353399, counted by A353398.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914, product A005361.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A324850 lists numbers divisible by the product of their prime indices.
Numbers divisible by their prime shadow:
- counted by A325702
- listed by A325755
- co-recursive version A325756
- nonprime recursive version A353389
- recursive version A353393, counted by A353426
Cf. A000005, A000961, A003586, A005117, A143773, A182850, A316428, A316438, A320325, A325131, A339095.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Select[Range[100],Times@@red/@primeMS[#]==red[#]&]

A181819(a(n)) = A353394(a(n)) = Product_i A181819(A112798(a(n),i)).

A317624 Number of integer partitions of n where all parts are > 1 and whose LCM is n.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 17, 1, 1, 1, 7, 1, 60, 1, 1, 1, 1, 1, 76, 1, 1, 1, 55, 1, 105, 1, 11, 10, 1, 1, 187, 1, 6, 1, 13, 1, 30, 1, 111, 1, 1, 1, 5043, 1, 1, 15, 1, 1, 230, 1, 17, 1, 242, 1, 4173, 1, 1, 12, 19, 1

Offset: 0

Gus Wiseman, Aug 01 2018

nonn

			The a(20) = 5 partitions are (20), (10,4,4,2), (10,4,2,2,2), (5,5,4,4,2), (5,5,4,2,2,2).
The a(45) = 10 partitions:
  (45),
  (15,15,9,3,3), (15,9,9,9,3),
  (15,9,9,3,3,3,3), (15,9,5,5,5,3,3), (9,9,9,5,5,5,3),
  (15,9,3,3,3,3,3,3,3), (9,9,5,5,5,3,3,3,3), (9,5,5,5,5,5,5,3,3),
  (9,5,5,5,3,3,3,3,3,3,3).
From _David A. Corneth_, Sep 08 2018: (Start)
Let sum(t) denote the sum of elements of a tuple t. The tuples t with distinct divisors of 45 that have lcm(t) = 45 and sum(t) <= 45 are {(45) and (3, 9, 15), (3, 5, 9, 15), (3, 5, 9), (5, 9), (9, 15), (5, 9, 15)}. For each such tuple t, find the number of partitions of 45 - s(t) into distinct parts of t.
For the tuple (45), there is 1 partition of 45 - 45 = 0 into parts with 45. That is: {()}.
For the tuple (3, 9, 15), there are 4 partitions of 45 - (3 + 9 + 15) = 18 into parts with 3, 9 and 15. They are {(3, 15), (9, 9), (3, 3, 3, 9), (3, 3, 3, 3, 3, 3)}.
For the tuple (3, 5, 9), there are 4 partitions of 45 - (3 + 5 + 9) = 28 into parts with 3, 5 and 9; they are {(5, 5, 9, 9), (3, 3, 3, 5, 5, 9), (3, 5, 5, 5, 5, 5), (3, 3, 3, 3, 3, 3, 5, 5)}.
For the tuple (3, 5, 9, 15), there is 1 partition of 45 - (3 + 5 + 9 + 15) = 13 into parts with 3, 5, 9 and 15. That is (3, 5, 5).
The other tuples, (5, 9), (9, 15), and (5, 9, 15); they give no extra tuples. That's because there is no solution to the Diophantine equation for 5x + 9y = 45 - (5 + 9), corresponding to the tuple (5, 9) with nonnegative x, y.
That also excludes (9, 15); if there is a solution for that, there would also be a solution for (5, 9). This could whittle down the number of seeds even further. Similarly, (5, 9, 15) gives no solution.
Therefore a(45) = 1 + 4 + 4 + 1 = 10.
(End)
In general, there are A318670(n) (<= A069626(n)) such seed sets of divisors where to start extending the partition from. (See the second PARI program which uses subroutine toplevel_starting_sets.) - _Antti Karttunen_, Sep 08 2018

Antti Karttunen, Table of n, a(n) for n = 0..359
Index entries for sequences related to lcm's

Cf. A000837, A018818, A066874, A067538, A069626, A074761, A074971, A143773, A259936, A281116, A285572, A290103, A305566, A316429, A316431, A316432, A316433, A318670.

Table[Length[Select[IntegerPartitions[n],And[Min@@#>=2,LCM@@#==n]&]],{n,30}]

strong_divisors_reversed(n) = vecsort(select(x -> (x>1), divisors(n)), , 4);
partitions_into_lcm(orgn,n,parts,from=1,m=1) = if(!n,(m==orgn),my(k = #parts, s=0); for(i=from,k,if(parts[i]<=n, s += partitions_into_lcm(orgn,n-parts[i],parts,i,lcm(m,parts[i])))); (s));
A317624(n) = if(n<=1,0,partitions_into_lcm(n,n,strong_divisors_reversed(n))); \\ Antti Karttunen, Sep 07 2018

strong_divisors_reversed(n) = vecsort(select(x -> (x>1), divisors(n)), , 4);
partitions_into(n,parts,from=1) = if(!n,1, if(#parts==from, (0==(n%parts[from])), my(s=0); for(i=from,#parts,if(parts[i]<=n, s += partitions_into(n-parts[i],parts,i))); (s)));
toplevel_starting_sets(orgn,n,parts,from=1,ss=List([])) = { my(k = #parts, s=0, newss); if(lcm(Vec(ss))==orgn,s += partitions_into(n,ss)); for(i=from,k,if(parts[i]<=n, newss = List(ss); listput(newss,parts[i]); s += toplevel_starting_sets(orgn,n-parts[i],parts,i+1,newss))); (s) };
A317624(n) = if(n<=1,0,toplevel_starting_sets(n,n,strong_divisors_reversed(n))); \\ Antti Karttunen, Sep 08-10 2018

A343657 Sum of number of divisors of x^y for each x >= 1, y >= 0, x + y = n.

Original entry on oeis.org

1, 2, 4, 7, 12, 18, 27, 39, 56, 77, 103, 134, 174, 223, 283, 356, 445, 547, 666, 802, 959, 1139, 1344, 1574, 1835, 2128, 2454, 2815, 3213, 3648, 4126, 4653, 5239, 5888, 6608, 7407, 8298, 9288, 10385, 11597, 12936, 14408, 16025, 17799, 19746, 21882, 24221

Offset: 1

Gus Wiseman, Apr 29 2021

nonn

			The a(7) = 27 divisors:
  1  32  81  64  25  6  1
     16  27  32  5   3
     8   9   16  1   2
     4   3   8       1
     2   1   4
     1       2
             1

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Antidiagonal row sums (row sums of the triangle) of A343656.
Dominated by A343661.
A000005(n) counts divisors of n.
A000312(n) = n^n.
A007318(n,k) counts k-sets of elements of {1..n}.
A009998(n,k) = n^k (as an array, offset 1).
A059481(n,k) counts k-multisets of elements of {1..n}.
A343658(n,k) counts k-multisets of divisors of n.
Cf. A000169, A000272, A002064, A002109, A048691, A062319, A066959, A143773, A146291, A176029, A251683, A282935, A326358, A327527, A334996.

Total/@Table[DivisorSigma[0,k^(n-k)],{n,30},{k,n}]

a(n) = Sum_{k=1..n} A000005(k^(n-k)).

A316440 Number of integer partitions of n such that every submultiset has an integer average.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 2, 7, 5, 8, 2, 13, 2, 10, 10, 14, 2, 20, 2, 17, 15, 14, 2, 32, 3, 16, 22, 25, 2, 40, 2, 27, 30, 20, 4, 58, 2, 22, 40, 40, 2, 64, 2, 40, 53, 26, 2, 93, 3, 30, 64, 54, 2, 94, 4, 58, 78, 32, 2, 138, 2, 34, 96, 75, 10, 131, 2, 76, 111, 48, 2, 192, 2, 40, 138, 99

Offset: 0

Gus Wiseman, Jul 03 2018

nonn

			The a(12) = 13 partitions:
  (12),
  (6,6), (7,5), (8,4), (9,3), (10,2), (11,1),
  (4,4,4), (6,4,2), (8,2,2),
  (3,3,3,3),
  (2,2,2,2,2,2),
  (1,1,1,1,1,1,1,1,1,1,1,1).

Max Alekseyev, Table of n, a(n) for n = 0..500

Cf. A067538, A074761, A122768, A143773, A237984, A298423, A316313, A316314.

Table[Length[Select[IntegerPartitions[n],And@@IntegerQ/@Mean/@Union[Rest[Subsets[#]]]&]],{n,20}]

For a prime p, a(p) = 2. - Max Alekseyev, Sep 02 2023

a(0) prepended and more terms added by Max Alekseyev, Sep 02 2023

cp's OEIS Frontend

A319333 Heinz numbers of integer partitions whose sum is equal to their LCM.

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A340851 Number of factorizations of n such that every factor is a divisor of the number of factors.

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A340853 Number of factorizations of n such that every factor is a multiple of the number of factors.

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A319567 Product of y divided by the GCD of y to the power of the length of y, where y is the integer partition with Heinz number n.

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A340693 Number of integer partitions of n where each part is a divisor of the number of parts.

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A343662 Irregular triangle read by rows where T(n,k) is the number of strict length k chains of divisors of n, 0 <= k <= Omega(n) + 1.

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A353395 Numbers k such that the prime shadow of k equals the product of prime shadows of the prime indices of k.

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A317624 Number of integer partitions of n where all parts are > 1 and whose LCM is n.

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PARI

PARI

A343657 Sum of number of divisors of x^y for each x >= 1, y >= 0, x + y = n.

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