A143773 -id:A143773 - cp's OEIS frontend

A143862 Number of compositions of n such that every part is divisible by number of parts.

1, 1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 9, 1, 7, 7, 9, 1, 19, 1, 14, 16, 11, 1, 43, 2, 13, 29, 34, 1, 56, 1, 51, 46, 17, 16, 130, 1, 19, 67, 139, 1, 105, 1, 142, 162, 23, 1, 315, 2, 151, 121, 246, 1, 219, 211, 321, 154, 29, 1, 1021, 1, 31, 219, 488, 496, 495, 1, 594, 232, 834, 1, 1439, 1

Offset: 0

Vladeta Jovovic, Sep 03 2008

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A143773, A261605.

{a(n) = if(n==0,1, sumdiv(n,d, binomial(n/d-1,d-1) ))}
for(n=0,50, print1(a(n),", ")) \\ Paul D. Hanna, Apr 25 2018

G.f.: Sum_{k>=0} x^(k^2) / (1 - x^k)^k.
G.f.: 1 + Sum_{n>=1} (1 + x^n)^(n-1) * x^n. - Paul D. Hanna, Jul 09 2019
a(n) = Sum_{d|n} binomial(n/d-1, d-1) for n>0 with a(0) = 1. - Paul D. Hanna, Apr 25 2018
G.f.: 1 + Sum_{n>=1} (x^n/(1-x^n))^n (conjecture). - Joerg Arndt, Jan 04 2024
For prime p, a(p) = 1, a(2*p) = p and a(p^2) = 2. - Peter Bala, Mar 02 2025

More terms from Franklin T. Adams-Watters, Apr 09 2009

A340828 Number of strict integer partitions of n whose maximum part is a multiple of their length.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 3, 2, 4, 5, 6, 6, 7, 8, 11, 10, 13, 17, 18, 21, 24, 27, 30, 35, 39, 46, 53, 61, 68, 79, 87, 97, 110, 123, 139, 157, 175, 196, 222, 247, 278, 312, 347, 385, 433, 476, 531, 586, 651, 720, 800, 883, 979, 1085, 1200, 1325, 1464, 1614, 1777

Offset: 1

Gus Wiseman, Feb 01 2021

nonn

			The a(1) = 1 through a(16) = 10 partitions (A..G = 10..16):
  1  2  3   4  5   6    7   8   9    A     B    C    D    E     F      G
        21     41  42   43  62  63   64    65   84   85   86    87     A6
                   321  61      81   82    83   A2   A3   A4    A5     C4
                                621  631   A1   642  C1   C2    C3     E2
                                     4321  632  651  643  653   E1     943
                                           641  921  652  932   654    952
                                                     931  941   942    961
                                                          8321  951    C31
                                                                C21    8431
                                                                8421   8521
                                                                54321

FindStat, St000010: The length of the partition.
FindStat, St000147: The largest part of an integer partition.
FindStat, St000784: The maximum of the length and the largest part of the integer partition.

Note: A-numbers of Heinz-number sequences are in parentheses below.
The non-strict version is A168659 (A340609/A340610).
A018818 counts partitions into divisors (A326841).
A047993 counts balanced partitions (A106529).
A064173 counts partitions of positive/negative rank (A340787/A340788).
A067538 counts partitions whose length/max divides sum (A316413/A326836).
A072233 counts partitions by sum and length, with strict case A008289.
A096401 counts strict partition with length equal to minimum.
A102627 counts strict partitions with length dividing sum.
A326842 counts partitions whose length and parts all divide sum (A326847).
A326850 counts strict partitions whose maximum part divides sum.
A326851 counts strict partitions with length and maximum dividing sum.
A340829 counts strict partitions with Heinz number divisible by sum.
A340830 counts strict partitions with all parts divisible by length.
Cf. A003114, A006141, A064174, A117409, A143773 (A316428), A200750, A326843 (A326837), A330950 (A324851).

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[Max@@#,Length[#]]&]],{n,30}]

A298423 Number of integer partitions of n such that the predecessor of each part is divisible by the number of parts.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 5, 2, 5, 4, 6, 2, 11, 2, 7, 8, 10, 2, 15, 2, 16, 11, 9, 2, 28, 7, 10, 14, 22, 2, 37, 2, 25, 18, 12, 17, 55, 2, 13, 23, 52, 2, 55, 2, 40, 51, 15, 2, 95, 13, 44, 34, 53, 2, 79, 37, 85, 41, 18, 2, 185, 2, 19, 80, 91, 54, 112, 2, 87, 56, 122, 2

Offset: 0

Gus Wiseman, Jan 19 2018

nonn

Note that n is automatically divisible by the number of parts.

			The a(9) = 4 partitions: (9), (441), (711), (111111111).

Cf. A000005, A000041, A067538, A143773, A280954, A298422, A298424, A298426.

Table[Length[Select[IntegerPartitions[n],Function[ptn,And@@(Divisible[#-1,Length[ptn]]&/@ptn)]]],{n,60}]

G.f.: Sum_{k>=0} x^k/Product_{i=1..k} (1-x^(k*i)).

A316429 Heinz numbers of integer partitions whose length is equal to their LCM.

Original entry on oeis.org

2, 6, 9, 20, 50, 56, 84, 125, 126, 176, 189, 196, 240, 294, 360, 416, 441, 540, 600, 624, 686, 810, 900, 936, 968, 1029, 1040, 1088, 1215, 1350, 1404, 1500, 1560, 2025, 2106, 2250, 2340, 2401, 2432, 2600, 2704, 3159, 3375, 3510, 3648, 3750, 3900, 4056, 5265

Offset: 1

Gus Wiseman, Jul 02 2018

A110295 is a subsequence.

			3750 is the Heinz number of (3,3,3,3,2,1), whose length and lcm are both 6.

David A. Corneth, Table of n, a(n) for n = 1..10822 (Terms <= 5 * 10^11)

Cf. A056239, A074761, A110295, A143773, A237984, A289508, A289509, A290103, A296150, A316413, A316428, A316430, A316431.

Select[Range[2,200],PrimeOmega[#]==LCM@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]]&]

heinz(n) = my(f=factor(n), pr=f[,1]~,exps=f[,2], res=vector(vecsum(exps)), t=0); for(i = 1, #pr, pr[i] = primepi(pr[i]); for(j=1, exps[i],t++; res[t] = pr[i])); res
is(n) = my(h = heinz(n)); lcm(h)==#h \\ David A. Corneth, Jul 05 2018

A340606 Numbers whose prime indices (A112798) are all divisors of the number of prime factors (A001222).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 16, 20, 24, 32, 36, 50, 54, 56, 64, 81, 84, 96, 125, 126, 128, 144, 160, 176, 189, 196, 216, 240, 256, 294, 324, 360, 384, 400, 416, 441, 486, 512, 540, 576, 600, 624, 686, 729, 810, 864, 896, 900, 936, 968, 1000, 1024, 1029, 1040, 1088, 1215

Offset: 1

Gus Wiseman, Jan 24 2021

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.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
   9: {2,2}
  16: {1,1,1,1}
  20: {1,1,3}
  24: {1,1,1,2}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  50: {1,3,3}
  54: {1,2,2,2}
  56: {1,1,1,4}
  64: {1,1,1,1,1,1}
  81: {2,2,2,2}
  84: {1,1,2,4}
  96: {1,1,1,1,1,2}

Note: Heinz numbers are given in parentheses below.
The reciprocal version is A143773 (A316428).
These partitions are counted by A340693.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
A003963 multiplies together the prime indices of n.
A018818 counts partitions of n into divisors of n (A326841).
A047993 counts balanced partitions (A106529).
A067538 counts partitions of n whose length divides n (A316413).
A056239 adds up the prime indices of n.
A061395 selects the maximum prime index.
A067538 counts partitions of n whose maximum divides n (A326836).
A072233 counts partitions by sum and length.
A112798 lists the prime indices of each positive integer.
A168659 = partitions whose length is divisible by their maximum (A340609).
A168659 = partitions whose maximum is divisible by their length (A340610).
A289509 lists numbers with relatively prime prime indices.
A326842 = partitions of n whose length and parts all divide n (A326847).
A326843 = partitions of n whose length and maximum both divide n (A326837).
A340852 have a factorization with factors dividing length.
Cf. A000720, A001222, A006141, A067539, A200750, A298423, A324925, A326149/A326155, A340608, A340827.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And@@IntegerQ/@(PrimeOmega[#]/primeMS[#])&]

A340830 Number of strict integer partitions of n such that every part is a multiple of the number of parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 4, 1, 4, 1, 6, 1, 5, 2, 6, 1, 8, 1, 7, 4, 7, 1, 12, 1, 8, 6, 9, 1, 16, 1, 10, 9, 11, 1, 21, 1, 12, 13, 12, 1, 28, 1, 13, 17, 16, 1, 33, 1, 19, 22, 15, 1, 45, 1, 16, 28, 25, 1, 47, 1, 28, 34, 18

Offset: 1

Gus Wiseman, Feb 02 2021

nonn

			The a(n) partitions for n = 1, 6, 10, 14, 18, 20, 24, 26, 30:
  1   6     10    14     18      20     24       26      30
      4,2   6,4   8,6    10,8    12,8   16,8     18,8    22,8
            8,2   10,4   12,6    14,6   18,6     20,6    24,6
                  12,2   14,4    16,4   20,4     22,4    26,4
                         16,2    18,2   22,2     24,2    28,2
                         9,6,3          14,10    14,12   16,14
                                        12,9,3   16,10   18,12
                                        15,6,3           20,10
                                                         15,9,6
                                                         18,9,3
                                                         21,6,3
                                                         15,12,3

Note: A-numbers of Heinz-number sequences are in parentheses below.
The non-strict case is A143773 (A316428).
The case where length divides sum also is A340827.
The version for factorizations is A340851.
Factorization of this type are counted by A340853.
A018818 counts partitions into divisors (A326841).
A047993 counts balanced partitions (A106529).
A067538 counts partitions whose length/max divide sum (A316413/A326836).
A072233 counts partitions by sum and length, with strict case A008289.
A102627 counts strict partitions whose length divides sum.
A326850 counts strict partitions whose maximum part divides sum.
A326851 counts strict partitions with length and maximum dividing sum.
A340828 counts strict partitions with length divisible by maximum.
A340829 counts strict partitions with Heinz number divisible by sum.
Cf. A114638, A168659, A326641, A326843 (A326837), A326849, A326852 (A326838), A330950 (A324851), A340852.

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

a(n) = Sum_{d|n} A008289(n/d, d).

A316430 Heinz numbers of integer partitions whose length is equal to the GCD of all the parts.

Original entry on oeis.org

1, 2, 9, 21, 39, 57, 87, 91, 111, 125, 129, 159, 183, 203, 213, 237, 247, 267, 301, 303, 321, 325, 339, 377, 393, 417, 427, 453, 489, 519, 543, 551, 553, 559, 575, 579, 597, 669, 687, 689, 707, 717, 753, 789, 791, 813, 817, 843, 845, 879, 923, 925, 933, 951, 973

Offset: 1

Gus Wiseman, Jul 02 2018

nonn

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

2 is the only even term in the sequence. 3k is in the sequence if and only if k is in A031215. 5k is in the sequence if and only if k = pq with p and q in A031336.

			Sequence of integer partitions whose length is equal to their GCD begins: (), (1), (2,2), (4,2), (6,2), (8,2), (10,2), (6,4), (12,2), (3,3,3), (14,2), (16,2), (18,2), (10,4), (20,2), (22,2), (8,6), (24,2), (14,4), (26,2), (28,2), (6,3,3).

Subsequence of A004280.

Cf. A056239, A067538, A074761, A143773, A289508, A289509, A296150, A316413, A316431, A316432, A316433, A031215, A031336.

Select[Range[200],PrimeOmega[#]==GCD@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]]&]

is(n,f=factor(n))=gcd(apply(primepi,f[,1]))==vecsum(f[,2]) \\ Charles R Greathouse IV, Jul 25 2024

a(n) << n log^2 n, can this be improved? - Charles R Greathouse IV, Jul 25 2024

A340827 Number of strict integer partitions of n into divisors of n whose length also divides n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 5, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 18, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 17, 1, 1, 1, 1, 1, 14, 1, 1, 1, 1, 1, 12, 1, 1, 1, 3, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Feb 01 2021

nonn

The first element not in A326715 that is however a Heinz number of these partitions is 273.

			The a(n) partitions for n = 6, 12, 24, 90, 84:
  6       12        24            90                      84
  3,2,1   6,4,2     12,8,4        45,30,15                42,28,14
          6,3,2,1   12,6,4,2      45,30,9,5,1             42,21,14,7
                    12,8,3,1      45,18,15,9,3            42,28,12,2
                    8,6,4,3,2,1   45,30,10,3,2            42,28,6,4,3,1
                                  45,18,15,10,2           42,28,7,4,2,1
                                  45,30,6,5,3,1           42,14,12,7,6,3
                                  45,30,9,3,2,1           42,21,12,4,3,2
                                  45,15,10,9,6,5          42,21,12,6,2,1
                                  45,18,10,9,5,3          42,21,14,4,2,1
                                  45,18,10,9,6,2          28,21,14,12,6,3
                                  45,18,15,6,5,1          28,21,14,12,7,2
                                  45,18,15,9,2,1          42,21,7,6,4,3,1
                                  30,18,15,10,6,5,3,2,1   42,14,12,7,4,3,2
                                                          42,14,12,7,6,2,1
                                                          28,21,14,12,4,3,2
                                                          28,21,14,12,6,2,1

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 2519 terms from Antti Karttunen)
Antti Karttunen, Scheme program for computing this sequence

Note: A-numbers of Heinz-number sequences are in parentheses below.
The non-strict case is A326842 (A326847).
A018818 = partitions using divisors (A326841).
A047993 = balanced partitions (A106529).
A067538 = partitions whose length/maximum divides sum (A316413/A326836).
A072233 = partitions by sum and length, with strict case A008289.
A102627 = strict partitions whose length divides sum.
A326850 = strict partitions whose maximum part divides sum.
A326851 = strict partitions w/ length and max dividing sum.
A340828 = strict partitions w/ length divisible by max.
A340829 = strict partitions w/ Heinz number divisible by sum.
A340830 = strict partitions w/ parts divisible by length.
Cf. A064173, A143773 (A316428), A168659, A326843 (A326837), A326852, A330950 (A324851), A340851, A340853.

Table[Length[Select[IntegerPartitions[n,All,Divisors[n]],UnsameQ@@#&&Divisible[n,Length[#]]&]],{n,30}]

A340827(n, divsleft=List(divisors(n)), rest=n, len=0) = if(rest<=0, !rest && !(n%len), my(s=0, d); forstep(i=#divsleft, 1, -1, d = divsleft[i]; listpop(divsleft,i); if(rest>=d, s += A340827(n, divsleft, rest-d, 1+len))); (s)); \\ Antti Karttunen, Feb 22 2023

;; See the Links-section. - Antti Karttunen, Feb 22 2023

Data section extended up to a(105) by Antti Karttunen, Feb 22 2023

A353398 Number of integer partitions of n where the product of multiplicities equals the product of prime shadows of the parts.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 2, 1, 2, 1, 2, 6, 5, 4, 4, 6, 6, 8, 8, 13, 16, 13, 16, 18, 16, 20, 21, 27, 30, 27, 33, 41, 44, 51, 48, 58, 61, 66, 66, 74, 83, 86, 99, 102, 111, 115, 126, 137, 147, 156

Offset: 0

Gus Wiseman, May 17 2022

nonn

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 a(8) = 1 through a(14) = 4 partitions (A = 10, B = 11):
  3311  711     61111  521111   5511      B11       A1111
        321111         3221111  9111      721111    731111
                                531111    811111    33221111
                                3321111   5221111   422111111
                                22221111  43111111
                                42111111

The LHS (product of multiplicities) is A005361, counted by A266477.
The RHS (product of prime shadows) is A353394, first appearances A353397.
A related comparison is A353396, ranked by A353395.
These partitions are ranked by A353399.
A001222 counts prime factors with multiplicity, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A325131 lists numbers relatively prime to their prime shadow.
A325755 lists numbers divisible by their prime shadow, counted by A325702.
A339095 counts partitions by product (or factorizations by sum).
Cf. A002033, A003963, A143773, A182850, A239455, A324850, A325756, A353393, A353426.

red[n_]:=If[n==1,1,Times@@Prime/@Last/@FactorInteger[n]];
Table[Length[Select[IntegerPartitions[n],Times@@red/@#==Times@@Length/@Split[#]&]],{n,0,30}]

A316433 Number of integer partitions of n whose length is equal to the LCM of all parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 2, 1, 4, 3, 4, 4, 8, 5, 7, 8, 10, 8, 13, 13, 20, 18, 25, 25, 36, 34, 48, 52, 64, 64, 85, 85, 108, 106, 129, 133, 160, 158, 189, 194, 229, 228, 276, 279, 332, 336, 394, 402, 476, 489, 572, 599, 699, 728, 845, 889, 1032, 1094, 1251, 1332, 1523

Offset: 1

Gus Wiseman, Jul 02 2018

nonn

			The a(13) = 8 partitions are (4441), (55111), (322222), (332221), (333211), (622111), (631111), (7111111).

Cf. A067538, A074761, A143773, A285572, A289508, A290103, A305566, A316429, A316430, A316431, A316432.

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

a(n) = {my(nb = 0); forpart(p=n, if (lcm(Vec(p))==#p, nb++);); nb;} \\ Michel Marcus, Jul 03 2018

cp's OEIS Frontend

A143862 Number of compositions of n such that every part is divisible by number of parts.

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A340828 Number of strict integer partitions of n whose maximum part is a multiple of their length.

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A298423 Number of integer partitions of n such that the predecessor of each part is divisible by the number of parts.

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A316429 Heinz numbers of integer partitions whose length is equal to their LCM.

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A340606 Numbers whose prime indices (A112798) are all divisors of the number of prime factors (A001222).

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A340830 Number of strict integer partitions of n such that every part is a multiple of the number of parts.

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A316430 Heinz numbers of integer partitions whose length is equal to the GCD of all the parts.

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A340827 Number of strict integer partitions of n into divisors of n whose length also divides n.

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