A298423 -id:A298423 - cp's OEIS frontend

A316413 Heinz numbers of integer partitions whose length divides their sum.

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 34, 37, 39, 41, 43, 46, 47, 49, 53, 55, 57, 59, 61, 62, 64, 67, 68, 71, 73, 78, 79, 81, 82, 83, 84, 85, 87, 88, 89, 90, 91, 94, 97, 98, 99, 100, 101, 103, 105, 107, 109, 110

Offset: 1

Gus Wiseman, Jul 02 2018

nonn

In other words, partitions whose average is an integer.

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

			Sequence of partitions whose length divides their sum begins (1), (2), (11), (3), (4), (111), (22), (31), (5), (6), (1111), (7), (8), (42), (51), (9), (33), (222), (411).

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1327 terms from R. J. Mathar)

Cf. A056239, A067538, A074761, A143773, A237984, A289508, A289509, A290103, A296150, A298423, A316428, A316431.

isA326413 := proc(n)
    psigsu := A056239(n) ;
    psigle := numtheory[bigomega](n) ;
    if modp(psigsu,psigle) = 0 then
        true;
    else
        false;
    end if;
end proc:
n := 1:
for i from 2 to 3000 do
    if isA326413(i) then
        printf("%d %d\n",n,i);
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Aug 09 2019
# second Maple program:
q:= n-> (l-> nops(l)>0 and irem(add(i, i=l), nops(l))=0)(map
        (i-> numtheory[pi](i[1])$i[2], ifactors(n)[2])):
select(q, [$1..110])[];  # Alois P. Heinz, Nov 19 2021

Select[Range[2,100],Divisible[Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]],PrimeOmega[#]]&]

A067538 Number of partitions of n in which the number of parts divides n.

Original entry on oeis.org

1, 2, 2, 4, 2, 8, 2, 11, 9, 14, 2, 46, 2, 24, 51, 66, 2, 126, 2, 202, 144, 69, 2, 632, 194, 116, 381, 756, 2, 1707, 2, 1417, 956, 316, 2043, 5295, 2, 511, 2293, 9151, 2, 10278, 2, 8409, 14671, 1280, 2, 36901, 8035, 21524, 11614, 25639, 2, 53138, 39810, 85004

Offset: 1

Naohiro Nomoto, Jan 27 2002

Also sum of p(n,d) over the divisors d of n, where p(n,m) is the count of partitions of n in exactly m parts. - Wouter Meeussen, Jun 07 2009
From Gus Wiseman, Sep 24 2019: (Start)
Also the number of integer partitions of n whose maximum part divides n. The Heinz numbers of these partitions are given by A326836. For example, the a(1) = 1 through a(8) = 11 partitions are:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                    (211)            (222)                (422)
                    (1111)           (321)                (431)
                                     (2211)               (2222)
                                     (3111)               (4211)
                                     (21111)              (22211)
                                     (111111)             (41111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)
(End)

			a(3)=2 because 3 is a prime; a(4)=4 because the five partitions of 4 are {4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}, and the number of parts in each of them divides 4 except for {2, 1, 1}.
From _Gus Wiseman_, Sep 24 2019: (Start)
The a(1) = 1 through a(8) = 11 partitions whose length divides their sum are the following. The Heinz numbers of these partitions are given by A316413.
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                    (31)             (42)                 (53)
                    (1111)           (51)                 (62)
                                     (222)                (71)
                                     (321)                (2222)
                                     (411)                (3221)
                                     (111111)             (3311)
                                                          (4211)
                                                          (5111)
                                                          (11111111)
(End)

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..500 from Wouter Meeussen, n = 501..1000 from Alois P. Heinz, n = 1001..5000 from David A. Corneth)
Eric W. Weisstein, Partition Function P
Wikipedia, Integer Partition

Cf. A000005, A000041, A143773, A298422, A298423, A298426.
The strict case is A102627.
Partitions with integer geometric mean are A067539.
Cf. A018818, A102627, A316413, A326622, A326836, A326843, A326850.

Do[p = IntegerPartitions[n]; l = Length[p]; c = 0; k = 1; While[k < l + 1, If[ IntegerQ[ n/Length[ p[[k]] ]], c++ ]; k++ ]; Print[c], {n, 1, 57}, All]
p[n_,k_]:=p[n,k]=p[n-1,k-1]+p[n-k,k];p[n_,k_]:=0/;k>n;p[n_,n_]:=1;p[n_,0]:=0
Table[Plus @@ (p[n,# ]&/ @ Divisors[n]),{n,36}] (* Wouter Meeussen, Jun 07 2009 *)
Table[Count[IntegerPartitions[n], q_ /; IntegerQ[Mean[q]]], {n, 50}]  (*Clark Kimberling, Apr 23 2019 *)

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

# uses A008284_T
from sympy import divisors
def A067538(n): return sum(A008284_T(n,d) for d in divisors(n,generator=True)) # Chai Wah Wu, Sep 21 2023

a(p) = 2 for all primes p.

Extended by Robert G. Wilson v, Oct 16 2002

A143773 Number of partitions of n such that every part is divisible by number of parts.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 5, 1, 4, 3, 6, 1, 8, 1, 7, 5, 6, 1, 14, 2, 7, 8, 11, 1, 17, 1, 14, 11, 9, 3, 29, 1, 10, 15, 23, 1, 28, 1, 23, 25, 12, 1, 51, 2, 20, 25, 32, 1, 44, 11, 39, 31, 15, 1, 94, 1, 16, 40, 52, 19, 64, 1, 57, 45, 44, 1, 126, 1, 19, 83, 74, 6, 90, 1, 124, 63, 21, 1, 186

Offset: 1

Vladeta Jovovic, Aug 31 2008

			The a(18) = 8 partitions are (18), (10 8), (12 6), (14 4), (16 2), (6 6 6), (9 6 3), (12 3 3). - _Gus Wiseman_, Jan 26 2018

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

Cf. A000005, A000041, A067538, A143862, A298422, A298423, A298426.

m = 100;
gf = Sum[x^(k^2)/Product[1-x^(k*i), {i, 1, k}], {k, 1, Sqrt[m]//Ceiling}];
CoefficientList[gf + O[x]^m, x] // Rest (* Jean-François Alcover, May 13 2019 *)

Vec(sum(k=1,20,x^(k^2)/prod(i=1,k,1-x^(k*i)+O(x^400)))) \\ Max Alekseyev, May 03 2009

G.f.: Sum(x^(k^2)/Product(1-x^(k*i), i=1..k), k=1..infinity).

For prime p, a(p) = 1 and a(p^2) = 2. For odd prime p, a(2*p) = (p + 1)/2. - Peter Bala, Mar 03 2025

More terms from Max Alekseyev, May 03 2009

A298422 Number of rooted trees with n nodes in which all positive outdegrees are the same.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 5, 2, 6, 4, 9, 2, 20, 2, 26, 12, 53, 2, 120, 2, 223, 43, 454, 2, 1100, 11, 2182, 215, 4902, 2, 11446, 2, 24744, 1242, 56014, 58, 131258, 2, 293550, 7643, 676928, 2, 1582686, 2, 3627780, 49155, 8436382, 2, 19809464, 50, 46027323, 321202

Offset: 1

Gus Wiseman, Jan 19 2018

nonn

Row sums of A298426.

			The a(9) = 6 trees: ((((((((o)))))))), (o(o(o(oo)))), (o((oo)(oo))), ((oo)(o(oo))), (ooo(oooo)), (oooooooo).

Alois P. Heinz, Table of n, a(n) for n = 1..200

Cf. A000005, A000081, A000598, A001190, A001678, A003238, A004111, A008864, A032305, A067538, A111299, A124343, A143773, A289078, A289079, A295461, A298118, A298204, A298423, A298424, A298426.

srut[n_]:=srut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[srut/@c]]]/@Select[IntegerPartitions[n-1],Function[ptn,And@@(Divisible[#-1,Length[ptn]]&/@ptn)]],SameQ@@Length/@Cases[#,{},{0,Infinity}]&]];
Table[srut[n]//Length,{n,20}]

a(n) = 2 <=> n in {A008864}. - Alois P. Heinz, Jan 20 2018

a(44)-a(52) from Alois P. Heinz, Jan 20 2018

A349156 Number of integer partitions of n whose mean is not an integer.

Original entry on oeis.org

1, 0, 0, 1, 1, 5, 3, 13, 11, 21, 28, 54, 31, 99, 111, 125, 165, 295, 259, 488, 425, 648, 933, 1253, 943, 1764, 2320, 2629, 2962, 4563, 3897, 6840, 6932, 9187, 11994, 12840, 12682, 21635, 25504, 28892, 28187, 44581, 42896, 63259, 66766, 74463, 104278, 124752

Offset: 0

Gus Wiseman, Nov 14 2021

nonn

Equivalently, partitions whose length does not divide their sum.

By conjugation, also the number of integer partitions of n with greatest part not dividing n.

			The a(3) = 1 through a(8) = 11 partitions:
  (21)  (211)  (32)    (2211)   (43)      (332)
               (41)    (3111)   (52)      (422)
               (221)   (21111)  (61)      (431)
               (311)            (322)     (521)
               (2111)           (331)     (611)
                                (421)     (22211)
                                (511)     (32111)
                                (2221)    (41111)
                                (3211)    (221111)
                                (4111)    (311111)
                                (22111)   (2111111)
                                (31111)
                                (211111)

Below, "!" means either enumerative or set theoretical complement.
The version for nonempty subsets is !A051293.
The complement is counted by A067538, ranked by A316413.
The geometric version is !A067539, strict !A326625, ranked by !A326623.
The strict case is !A102627.
The version for prime factors is A175352, complement A078175.
The version for distinct prime factors is A176587, complement A078174.
The ordered version (compositions) is !A271654, ranked by !A096199.
The multiplicative version (factorizations) is !A326622, geometric !A326028.
The conjugate is ranked by !A326836.
The conjugate strict version is !A326850.
These partitions are ranked by A348551.
A000041 counts integer partitions.
A326567/A326568 give the mean of prime indices, conjugate A326839/A326840.
A236634 counts unbalanced partitions, complement of A047993.
A327472 counts partitions not containing their mean, complement of A237984.
Cf. A001700, A074761, A098743, A143773, A175397, A175761, A298423, A326027, A326641, A326842, A326849, A327778.

Table[Length[Select[IntegerPartitions[n],!IntegerQ[Mean[#]]&]],{n,0,30}]

a(n > 0) = A000041(n) - A067538(n).

A316428 Heinz numbers of integer partitions such that every part is divisible by the number of parts.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 29, 31, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 67, 71, 73, 79, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 125, 127, 129, 131, 133, 137, 139, 149, 151, 157, 159, 163, 167, 169, 173, 179, 181, 183, 191, 193, 197

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).

			93499 is the Heinz number of (12,8,8,4) and belongs to the sequence because each part is divisible by 4.
Sequence of partitions such that every part is divisible by the number of parts begins (1), (2), (3), (4), (2,2), (5), (6), (7), (8), (4,2), (9).

Cf. A056239, A067538, A074761, A143773, A237984, A289509, A296150, A298423, A316413.

Select[Range[200],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>Divisible[PrimePi[p],PrimeOmega[#]]]&]

A298426 Regular triangle where T(n,k) is number of k-ary rooted trees with n nodes.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 2, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 3, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 6, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 11, 4, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 23, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gus Wiseman, Jan 19 2018

Row sums are A298422.

			Triangle begins:
1
0  1
0  1  1
0  1  0  1
0  1  1  0  1
0  1  0  0  0  1
0  1  2  1  0  0  1
0  1  0  0  0  0  0  1
0  1  3  0  1  0  0  0  1
0  1  0  2  0  0  0  0  0  1
0  1  6  0  0  1  0  0  0  0  1
0  1  0  0  0  0  0  0  0  0  0  1
0  1  11 4  2  0  1  0  0  0  0  0  1
0  1  0  0  0  0  0  0  0  0  0  0  0  1
0  1  23 0  0  0  0  1  0  0  0  0  0  0  1
0  1  0  8  0  2  0  0  0  0  0  0  0  0  0  1

Alois P. Heinz, Rows n = 1..200, flattened

Cf. A000005, A000081, A000598, A001190, A001678, A003238, A004111, A067538, A143773, A289078, A289079, A295461, A298118, A298422, A298423, A298424.

nn=16;
arut[n_,k_]:=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[arut[#,k]&/@c]]]/@Select[IntegerPartitions[n-1],Length[#]===k&]]
Table[arut[n,k]//Length,{n,nn},{k,0,n-1}]

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[#])&]

A298424 Matula-Goebel numbers of rooted trees in which all positive outdegrees are the same.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 11, 14, 16, 31, 32, 49, 64, 76, 86, 127, 128, 256, 301, 424, 454, 512, 709, 722, 886, 1024, 1532, 1589, 1849, 2048, 2096, 3101, 3986, 4096, 5381, 6418, 6859, 8192, 9761, 9952, 11236, 13766, 13951, 14554, 16384, 19049, 21884, 22463, 23512

Offset: 1

Gus Wiseman, Jan 19 2018

nonn

			Sequence of trees begins:
1   o
2   (o)
3   ((o))
4   (oo)
5   (((o)))
8   (ooo)
11  ((((o))))
14  (o(oo))
16  (oooo)
31  (((((o)))))
32  (ooooo)
49  ((oo)(oo))
64  (oooooo)
76  (oo(ooo))
86  (o(o(oo)))
127 ((((((o))))))
128 (ooooooo)
256 (oooooooo)
301 ((oo)(o(oo)))
424 (ooo(oooo))
454 (o((oo)(oo)))
512 (ooooooooo)
709 (((((((o)))))))
722 (o(ooo)(ooo))
886 (o(o(o(oo))))

Cf. A000081, A000598, A001190, A003238, A007097, A111299, A214577, A276625, A298422, A298423, A298426.

MGtree[n_]:=If[n===1,{},MGtree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
soQ[n_]:=Or[n===1,SameQ@@Length/@Cases[MGtree[n],{},{0,Infinity}]];
Select[Range[1000],soQ]

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}]

cp's OEIS Frontend

A316413 Heinz numbers of integer partitions whose length divides their sum.

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A067538 Number of partitions of n in which the number of parts divides n.

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A143773 Number of partitions of n such that every part is divisible by number of parts.

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A298422 Number of rooted trees with n nodes in which all positive outdegrees are the same.

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A349156 Number of integer partitions of n whose mean is not an integer.

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A316428 Heinz numbers of integer partitions such that every part is divisible by the number of parts.

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A298426 Regular triangle where T(n,k) is number of k-ary rooted trees with n nodes.

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