A279787 -id:A279787 - cp's OEIS frontend

A319071 Number of integer partitions of n whose product of parts is a perfect power and whose parts all have the same number of prime factors, counted with multiplicity.

1, 0, 0, 0, 2, 0, 2, 0, 3, 2, 3, 0, 4, 1, 4, 3, 7, 1, 7, 1, 8, 6, 8, 0, 15, 5, 12, 6, 15, 4, 22, 4, 24, 12, 22, 8, 35, 7, 30, 16, 42, 9, 50, 9, 50, 30, 53, 7, 79, 22, 72, 33, 87, 21, 109, 26, 111, 55, 117, 24, 168, 40, 149, 65, 178, 59

Offset: 0

Gus Wiseman, Oct 10 2018

nonn

The positions of zeros appear to be A048278.

			The a(4) = 2 through a(16) = 7 integer partitions (G = 16):
  4   33   8     9    55     66      94  77       555     G
  22  222  44    333  3322   444         5522     33333   88
           2222       22222  3333        332222   333222  664
                             222222      2222222          4444
                                                          5533
                                                          333322
                                                          22222222

Cf. A003963, A048278, A064573, A279787, A305551, A319056, A319066, A319169, A320322, A320323.

Table[Length[Select[IntegerPartitions[n],And[GCD@@FactorInteger[Times@@#][[All,2]]>1,SameQ@@PrimeOmega/@#]&]],{n,30}]

A355385 Number of pairs (y, v) of integer partitions of n where the length of v equals the number of distinct parts in y.

Original entry on oeis.org

1, 1, 2, 3, 7, 12, 25, 43, 81, 141, 243, 409, 699, 1132, 1844, 2995, 4744, 7408, 11655, 17839, 27509, 41546, 62879, 93537, 139974, 205547, 302714, 440097, 640968, 921774, 1327538, 1891548, 2696635, 3809860, 5380257, 7540778, 10561566, 14687109, 20408170, 28183998, 38882009

Offset: 0

Gus Wiseman, Jul 02 2022

nonn

Also the number of composable pairs of integer partitions of n, where a partition is regarded as an arrow from (number of parts) to (number of distinct parts). Is there a nice choice of a composition operation making this into an associative category?

			The a(0) = 1 through a(5) = 10 pairs:
  ()()  (1)(1)  (2)(2)   (3)(3)    (4)(4)     (5)(5)
                (11)(2)  (21)(21)  (31)(31)   (41)(41)
                         (111)(3)  (31)(22)   (41)(32)
                                   (22)(4)    (32)(41)
                                   (211)(31)  (32)(32)
                                   (211)(22)  (311)(41)
                                   (1111)(4)  (311)(32)
                                              (221)(41)
                                              (221)(32)
                                              (2111)(41)
                                              (2111)(32)
                                              (11111)(5)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The inhomogeneous version with containment and multiplicity is A339006.
The inhomogeneous version with containment is A355383.
The inhomogeneous version with containment for compositions is A355384.
The version for compositions is A355388.
A001970 counts multiset partitions of partitions.
A063834 counts partitions of each part of a partition.
A323583 counts splittings of partitions.
Cf. A000009, A008284, A022811, A032020, A070933, A072706, A116608, A279787, A319910, A336135.

Table[Length[Select[Tuples[IntegerPartitions[n],2],Length[Union[#[[1]]]]==Length[#[[2]]]&]],{n,0,15}]

\\ P gives A008284 and R gives A116608 as g.f.'s.
P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
R(n,y) = {prod(k=1, n, 1 + y/(1 - x^k) - y + O(x*x^n))}
seq(n) = {my(g=Vec(P(n,y)), h=Vec(R(n,y))); vector(n+1, i, my(p=g[i], q=h[i]); sum(j=0, poldegree(q), polcoef(p,j)*polcoef(q,j)))} \\ Andrew Howroyd, Dec 31 2022

a(n) = Sum_{j >= 1} A116608(n,j) * A008284(n,j) for n > 0. - Andrew Howroyd, Dec 31 2022

Terms a(26) and beyond from Andrew Howroyd, Dec 31 2022

A358833 Number of rectangular twice-partitions of n of type (P,R,P).

Original entry on oeis.org

1, 1, 3, 4, 8, 8, 17, 16, 32, 34, 56, 57, 119, 102, 179, 199, 335, 298, 598, 491, 960, 925, 1441, 1256, 2966, 2026, 3726, 3800, 6488, 4566, 11726, 6843, 16176, 14109, 21824, 16688, 49507, 21638, 50286, 50394, 99408, 44584, 165129, 63262, 208853, 205109, 248150

Offset: 0

Gus Wiseman, Dec 04 2022

nonn

A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n, so these are twice-partitions of n into partitions with constant lengths and constant sums.

			The a(1) = 1 through a(5) = 8 twice-partitions:
  (1)  (2)     (3)        (4)           (5)
       (11)    (21)       (22)          (32)
       (1)(1)  (111)      (31)          (41)
               (1)(1)(1)  (211)         (221)
                          (1111)        (311)
                          (2)(2)        (2111)
                          (11)(11)      (11111)
                          (1)(1)(1)(1)  (1)(1)(1)(1)(1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences enumerating triangles of integer partitions

This is the rectangular case of A279787.
This is the case of A306319 with constant sums.
For distinct instead of constant lengths and sums we have A358832.
The version for multiset partitions of integer partitions is A358835.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A281145 counts same-trees.
Cf. A000041, A000219, A001970, A008284, A141199, A327908, A358823, A358831.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],SameQ@@Length/@#&&SameQ@@Total/@#&]],{n,0,10}]

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(u=Vec(P(n,y)-1)); concat([1], vector(n, n, sumdiv(n, d, my(p=u[n/d]); sum(j=1, n/d, polcoef(p, j, y)^d))))} \\ Andrew Howroyd, Dec 31 2022

a(n) = Sum_{d|n} Sum_{j=1..n/d} A008284(n/d, j)^d for n > 0. - Andrew Howroyd, Dec 31 2022

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2022

A374704 Number of ways to choose an integer partition of each part of an integer composition of n (A055887) such that the minima are identical.

Original entry on oeis.org

1, 1, 3, 6, 15, 31, 77, 171, 410, 957, 2275, 5370, 12795, 30366, 72307, 172071, 409875, 976155, 2325804, 5541230, 13204161, 31464226, 74980838, 178684715, 425830008, 1014816979, 2418489344, 5763712776, 13736075563, 32735874251, 78016456122, 185929792353, 443110675075

Offset: 0

Gus Wiseman, Aug 04 2024

nonn

			The a(0) = 1 through a(4) = 15 ways:
  ()  ((1))  ((2))      ((3))          ((4))
             ((1,1))    ((1,2))        ((1,3))
             ((1),(1))  ((1,1,1))      ((2,2))
                        ((1),(1,1))    ((1,1,2))
                        ((1,1),(1))    ((2),(2))
                        ((1),(1),(1))  ((1,1,1,1))
                                       ((1),(1,2))
                                       ((1,2),(1))
                                       ((1),(1,1,1))
                                       ((1,1),(1,1))
                                       ((1,1,1),(1))
                                       ((1),(1),(1,1))
                                       ((1),(1,1),(1))
                                       ((1,1),(1),(1))
                                       ((1),(1),(1),(1))

Andrew Howroyd, Table of n, a(n) for n = 0..1000

A variation for weakly increasing lengths is A141199.
For identical sums instead of minima we have A279787.
The case of reversed twice-partitions is A306319, distinct A358830.
For maxima instead of minima, or for unreversed partitions, we have A358905.
The strict case is A374686 (ranks A374685), maxima A374760 (ranks A374759).
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A055887 counts sequences of partitions with total sum n.
A281145 counts same-trees.
A319169 counts partitions with constant Omega, ranked by A320324.
A358911 counts compositions with constant Omega, distinct A358912.
Cf. A000041, A063834, A106356, A189076, A238343, A304969, A305551, A319066, A323429, A333213, A358833, A358835.

Table[Length[Select[Join@@Table[Tuples[IntegerPartitions/@y], {y,Join@@Permutations/@IntegerPartitions[n]}],SameQ@@Min/@#&]],{n,0,15}]

seq(n) = Vec(1 + sum(k=1, n, -1 + 1/(1 - x^k/prod(j=k, n-k, 1 - x^j, 1 + O(x^(n-k+1)))))) \\ Andrew Howroyd, Dec 29 2024

G.f.: 1 + Sum_{k>=1} (-1 + 1/(1 - x^k/Product_{j>=k} (1 - x^j))). - Andrew Howroyd, Dec 29 2024

a(16) onwards from Andrew Howroyd, Dec 29 2024

A320698 Numbers whose product of prime indices is a prime power (A246655).

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 12, 14, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 31, 34, 36, 38, 40, 41, 42, 44, 46, 48, 49, 50, 53, 54, 56, 57, 59, 62, 63, 67, 68, 72, 76, 80, 81, 82, 83, 84, 88, 92, 96, 97, 98, 100, 103, 106, 108, 109, 112, 114, 115, 118, 121, 124

Offset: 1

Gus Wiseman, Oct 19 2018

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.

Also numbers whose prime indices are all powers of a common prime number.

			The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (2), (3), (1,2), (4), (2,2), (1,3), (5), (1,1,2), (1,4), (7), (1,2,2), (8), (1,1,3), (2,4), (1,5), (9), (1,1,1,2), (3,3), (2,2,2), (1,1,4), (11), (1,7), (1,1,2,2), (1,8), (1,1,1,3), (13), (1,2,4), (1,1,5), (1,9), (1,1,1,1,2), (4,4), (1,3,3), (16), (1,2,2,2), (1,1,1,4), (2,8).

Index entries for sequences computed from indices in prime factorization

Cf. A000720, A000961, A001597, A003963, A047968, A056239, A064573, A112798, A246547, A246655, A279787, A320322, A320325, A320699, A320700.

Select[Range[100],PrimePowerQ[Times@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]^k]]&]

is(n) = my(f=factor(n)[, 1]~, p=1); for(k=1, #f, p=p*primepi(f[k])); isprimepower(p) \\ Felix Fröhlich, Oct 20 2018

A323531 Number of square multiset partitions of integer partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 5, 9, 12, 18, 24, 36, 48, 69, 97, 139, 196, 283, 402, 576, 819, 1161, 1635, 2301, 3209, 4469, 6193, 8571, 11812, 16291, 22404, 30850, 42414, 58393, 80305, 110578, 152091, 209308, 287686, 395352, 542413, 743603, 1017489, 1390510, 1896482

Offset: 0

Gus Wiseman, Jan 21 2019

nonn

A multiset partition is square if the number of parts is equal to the number of parts in each part.

			The a(3) = 1 through a(9) = 12 square multiset partitions:
  (3)  (4)       (5)       (6)       (7)       (8)       (9)
       (11)(11)  (21)(11)  (21)(21)  (22)(21)  (22)(22)  (32)(22)
                           (22)(11)  (31)(21)  (31)(22)  (32)(31)
                           (31)(11)  (32)(11)  (31)(31)  (33)(21)
                                     (41)(11)  (32)(21)  (41)(22)
                                               (33)(11)  (41)(31)
                                               (41)(21)  (42)(21)
                                               (42)(11)  (43)(11)
                                               (51)(11)  (51)(21)
                                                         (52)(11)
                                                         (61)(11)
                                                         (111)(111)(111)

Cf. A000219, A001970, A047968, A261049, A279787, A305551, A319066, A323580.

Table[Sum[Length[Union@@(Union[Sort/@Tuples[IntegerPartitions[#,{k}]&/@#]]&/@IntegerPartitions[n,{k}])],{k,Sqrt[n]}],{n,30}]

A371733 Maximal length of a factorization of n into factors > 1 all having the same sum of prime indices.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 5, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4

Offset: 1

Gus Wiseman, Apr 13 2024

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. Sum of prime indices is given by A056239.

Factorizations into factors all having the same sum of prime indices are counted by A321455.

			The factorizations of 588 of this type are (7*7*12), (21*28), (588), so a(588) = 3.
The factorizations of 900 of this type are (5*5*6*6), (9*10*10), (25*36), (30*30), (900), so a(900) = 4.

Positions of 1's are A321453, counted by A321451.
Positions of terms > 1 are A321454, counted by A321452.
Factorizations of this type are counted by A321455, different sums A321469.
For different sums instead of same sums we have A371734.
For set partitions of binary indices we have A371735.
A001055 counts factorizations.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A321142 and A371794 count non-biquanimous strict partitions.
A371789 counts non-quanimous sets, differences A371790.
A371796 counts quanimous sets, differences A371797.
Cf. A035470, A279787, A305551, A322794, A326515, A326518, A326534, A336137, A371783, A371791.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Table[Max[Length/@Select[facs[n],SameQ@@hwt/@#&]],{n,100}]

A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
all_have_same_sum_of_pis(facs) = if(!#facs, 1, (#Set(apply(A056239,facs)) == 1));
A371733(n, m=n, facs=List([])) = if(1==n, if(all_have_same_sum_of_pis(facs),#facs,0), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s = max(s, A371733(n/d, d, newfacs)))); (s)); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(108) by Antti Karttunen, Jan 20 2025

A371734 Maximal length of a factorization of n into factors > 1 all having different sums of prime indices.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 13 2024

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. Sum of prime indices is given by A056239.

Factorizations into factors > 1 all having different sums of prime indices are counted by A321469.

			The factorizations of 90 of this type are (2*3*15), (2*5*9), (2*45), (3*30), (5*18), (6*15), (90), so a(90) = 3.

For set partitions of binary indices we have A000120, same sums A371735.
Positions of 1's are A000430.
Positions of terms > 1 are A080257.
Factorizations of this type are counted by A321469, same sums A321455.
For same instead of different sums we have A371733.
A001055 counts factorizations.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
Cf. A035470, A279787, A305551, A321142, A322794, A326515, A326518, A326534, A336137, A371783, A371789, A371791.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Table[Max[Length/@Select[facs[n],UnsameQ@@hwt/@#&]],{n,100}]

A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
all_have_different_sum_of_pis(facs) = if(!#facs, 1, (#Set(apply(A056239,facs)) == #facs));
A371734(n, m=n, facs=List([])) = if(1==n, if(all_have_different_sum_of_pis(facs),#facs,0), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s = max(s,A371734(n/d, d, newfacs)))); (s)); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

A371735 Maximal length of a set partition of the binary indices of n into blocks all having the same sum.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Apr 14 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

If a(n) = k then the binary indices of n (row n of A048793) are k-quanimous (counted by A371783).

			The binary indices of 119 are {1,2,3,5,6,7}, and the set partitions into blocks with the same sum are:
  {{1,7},{2,6},{3,5}}
  {{1,5,6},{2,3,7}}
  {{1,2,3,6},{5,7}}
  {{1,2,3,5,6,7}}
So a(119) = 3.

Set partitions of this type are counted by A035470, A336137.
A version for factorizations is A371733.
Positions of 1's are A371738.
Positions of terms > 1 are A371784.
A001055 counts factorizations.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A048793 lists binary indices, length A000120, reverse A272020, sum A029931.
A070939 gives length of binary expansion.
A321452 counts quanimous partitions, ranks A321454.
A326031 gives weight of the set-system with BII-number n.
A371783 counts k-quanimous partitions.
A371789 counts non-quanimous sets, differences A371790.
A371796 counts quanimous sets, differences A371797.
Cf. A006827, A038041, A096111, A279787, A305551, A321451, A321455, A322794, A326534, A371731, A371734.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Max[Length/@Select[sps[bix[n]],SameQ@@Total/@#&]],{n,0,100}]

A306318 Number of square twice-partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 5, 10, 12, 19, 24, 39, 49, 73, 104, 151, 212, 317, 443, 638, 936, 1296, 1841, 2635, 3641, 5069, 7176, 9884, 13614, 19113, 26162, 36603, 50405, 70153, 96176, 135388, 184753, 257882, 353587, 494653, 671992, 934905, 1272195, 1762979, 2389255

Offset: 0

Gus Wiseman, Feb 07 2019

nonn

A twice partition of n is a sequence of integer partitions, one of each part in an integer partition of n. It is square if the number of parts is equal to the number of parts in each part.

			The a(10) = 19 square twice-partitions:
  ((ten))  ((32)(32))  ((211)(111)(111))
           ((32)(41))
           ((33)(22))
           ((33)(31))
           ((41)(32))
           ((41)(41))
           ((42)(22))
           ((42)(31))
           ((43)(21))
           ((44)(11))
           ((51)(22))
           ((51)(31))
           ((52)(21))
           ((53)(11))
           ((61)(21))
           ((62)(11))
           ((71)(11))

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000219, A001970, A063834 (twice-partitions), A089299 (square plane partitions), A279787, A305551, A306017, A306319 (rectangular twice-partitions), A319066, A323429, A323531 (square partitions of partitions).

Table[Sum[Length[Union@@(Tuples[IntegerPartitions[#,{k}]&/@#]&/@IntegerPartitions[n,{k}])],{k,0,Sqrt[n]}],{n,0,20}]

cp's OEIS Frontend

A319071 Number of integer partitions of n whose product of parts is a perfect power and whose parts all have the same number of prime factors, counted with multiplicity.

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A355385 Number of pairs (y, v) of integer partitions of n where the length of v equals the number of distinct parts in y.

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A358833 Number of rectangular twice-partitions of n of type (P,R,P).

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A374704 Number of ways to choose an integer partition of each part of an integer composition of n (A055887) such that the minima are identical.

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A320698 Numbers whose product of prime indices is a prime power (A246655).

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A323531 Number of square multiset partitions of integer partitions of n.

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A371733 Maximal length of a factorization of n into factors > 1 all having the same sum of prime indices.

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