A320322 -id:A320322 - cp's OEIS frontend

A322528 Number of integer partitions of n whose parts all have the same number of prime factors (counted with multiplicity) and whose product of parts is a power of a squarefree number (A072774).

1, 1, 2, 2, 3, 3, 4, 3, 5, 4, 7, 2, 7, 4, 7, 7, 9, 3, 10, 5, 12, 9, 8, 6, 14, 10, 12, 10, 14, 11, 20, 13, 18, 13, 16, 16, 25, 16, 19, 20, 26, 18, 30, 19, 27, 26, 27, 22, 38, 30, 37, 28, 38, 32, 43, 37, 46, 40, 47, 40, 66, 49, 58, 56, 64, 56, 73, 58, 76, 70, 85

Offset: 0

Gus Wiseman, Dec 14 2018

nonn

			The a(1) = 1 through a(8) = 5 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (52)       (44)
                    (1111)  (11111)  (222)     (1111111)  (53)
                                     (111111)             (2222)
                                                          (11111111)

Cf. A002865, A003963, A005117, A064573 , A072774, A295193, A302505, A306017, A319169, A320322, A322526, A322527, A322529, A322530, A322531.

Table[Length[Select[IntegerPartitions[n],And[SameQ@@PrimeOmega/@#,SameQ@@Last/@FactorInteger[Times@@#]]&]],{n,30}]

More terms from Alois P. Heinz, Dec 14 2018

A322529 Number of integer partitions of n whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 2, 3, 2, 2, 4, 2, 3, 3, 4, 4, 4, 3, 5, 4, 5, 6, 6, 6, 6, 6, 8, 6, 7, 9, 8, 11, 8, 11, 11, 11, 12, 13, 13, 15, 13, 17, 17, 18, 18, 17, 20, 22, 21, 24, 24, 24, 26, 29, 28, 33, 30, 35, 34, 38, 38, 45, 42, 43, 45, 48, 52, 54, 55, 59, 59, 65, 65, 72, 73

Offset: 0

Gus Wiseman, Dec 14 2018

nonn

Such a partition must be strict (unless it is all 1's) and its parts must also be squarefree.

			The a(30) = 8 integer partitions:
  (30),
  (17,13),(19,11),(23,7),
  (17,11,2),(23,5,2),
  (13,7,5,3,2),
  (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1).

Lucas A. Brown, Table of n, a(n) for n = 0..133
Lucas A. Brown, Python program.

Cf. A002865, A003963, A005117, A038041, A064573, A073576, A302505, A319056, A320322, A321717, A321718, A322526, A322527, A322528, A322530, A322531.

Table[Length[Select[IntegerPartitions[n],And[SameQ@@PrimeOmega/@#,SquareFreeQ[Times@@#]]&]],{n,30}]

a(51)-a(69) from Jinyuan Wang, Jun 27 2020

a(70) onwards from Lucas A. Brown, Aug 17 2024

A325045 Number of factorizations of n whose conjugate as an integer partition has no ones.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 27 2019

nonn

After a(1) = 1, a(n) is the number of factorizations of n with at least two factors, the largest two of which are equal.

			The initial terms count the following factorizations:
    1: {}
    4: 2*2
    8: 2*2*2
    9: 3*3
   16: 2*2*2*2
   16: 4*4
   18: 2*3*3
   25: 5*5
   27: 3*3*3
   32: 2*2*2*2*2
   32: 2*4*4
   36: 2*2*3*3
   36: 6*6
   48: 3*4*4
   49: 7*7
   50: 2*5*5
   54: 2*3*3*3
   64: 2*2*2*2*2*2
   64: 2*2*4*4
   64: 4*4*4
   64: 8*8

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001055, A001222, A002865, A096276, A114324, A122111, A318950, A319005, A319916, A320322, A321648, A325039, A353645 [= a(n^2)].

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[facs[n],FreeQ[conj[#],1]&]],{n,1,100}]

A325045(n, m=n, facs=List([])) = if(1==n, (0==#facs || (#facs>=2 && facs[1]==facs[2])), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A325045(n/d, d, newfacs))); (s)); \\ Antti Karttunen, May 03 2022

More terms from Antti Karttunen, May 03 2022

A322453 Number of factorizations of n into factors > 1 using only primes and perfect powers.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 7, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 7, 1, 2, 2, 5, 1, 1, 1, 3, 1

Offset: 1

Gus Wiseman, Dec 09 2018

nonn

First differs from A000688 at a(36) = 5, A000688(36) = 4.

Terms in this sequence only depend on the prime signature of n. - David A. Corneth, Dec 26 2018

			The a(144) = 13 factorizations:
  (144),
  (4*36), (9*16),
  (2*2*36), (2*8*9), (3*3*16), (4*4*9),
  (2*2*4*9), (2*3*3*8), (3*3*4*4),
  (2*2*2*2*9), (2*2*3*3*4),
  (2*2*2*2*3*3).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A000688, A000961, A001055, A001597, A025487, A050336, A284696, A294068, A320322, A322452.

perpowQ[n_]:=GCD@@FactorInteger[n][[All,2]]>1;
pfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[pfacs[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],Or[PrimeQ[#],perpowQ[#]]&]}]];
Table[Length[pfacs[n]],{n,100}]

A322453(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&(ispower(d)||isprime(d)), s += A322453(n/d, d))); (s)); \\ Antti Karttunen, Dec 26 2018

More terms from Antti Karttunen, Dec 24 2018

A339452 Number of compositions (ordered partitions) of n into distinct parts such that the geometric mean of the parts is an integer.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 7, 1, 1, 5, 1, 1, 9, 7, 3, 1, 3, 1, 7, 11, 13, 1, 7, 1, 11, 35, 25, 31, 27, 5, 157, 1, 31, 131, 39, 31, 33, 37, 183, 179, 135, 157, 7, 265, 3, 871, 187, 865, 259, 879, 867, 179, 1593, 6073, 1593, 271, 5995, 149, 6661, 2411, 1509, 997, 1045, 5887

Offset: 1

Ilya Gutkovskiy, Dec 05 2020

nonn

			a(10) = 5 because we have [10], [9, 1], [1, 9], [8, 2] and [2, 8].

Eric W. Weisstein's World of Mathematics, Geometric Mean
Index entries for sequences related to compositions

For partitions we have A326625, non-strict A067539 (ranked by A326623).
The version for subsets is A326027.
For arithmetic mean we have A339175, non-strict A271654.
The non-strict case is counted by A357710, ranked by A357490.
A032020 counts strict compositions.
A067538 counts partitions with integer average.
A078175 lists numbers whose prime factors have integer average.
A320322 counts partitions whose product is a perfect power.
Cf. A051293, A078174, A096199, A102627, A326622, A326624, A326028, A326641, A326645, A335405.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&IntegerQ[GeometricMean[#]]&]],{n,0,15}] (* Gus Wiseman, Oct 30 2022 *)

A320699 Numbers whose product of prime indices is a nonprime prime power (A246547).

Original entry on oeis.org

7, 9, 14, 18, 19, 21, 23, 25, 27, 28, 36, 38, 42, 46, 49, 50, 53, 54, 56, 57, 63, 72, 76, 81, 84, 92, 97, 98, 100, 103, 106, 108, 112, 114, 115, 121, 125, 126, 131, 133, 144, 147, 152, 159, 162, 168, 171, 184, 189, 194, 196, 200, 206, 212, 216, 224, 227, 228

Offset: 1

Gus Wiseman, Oct 19 2018

nonn

First differs from A320325 at a(43) = 152, A320325(43) = 151.

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 all integer partitions whose Heinz numbers belong to the sequence begins: (4), (2,2), (4,1), (2,2,1), (8), (4,2), (9), (3,3), (2,2,2), (4,1,1), (2,2,1,1), (8,1), (4,2,1), (9,1), (4,4), (3,3,1), (16), (2,2,2,1), (4,1,1,1), (8,2), (4,2,2), (2,2,1,1,1), (8,1,1), (2,2,2,2), (4,2,1,1), (9,1,1), (25), (4,4,1), (3,3,1,1).

Index entries for sequences computed from indices in prime factorization

Cf. A000720, A000961, A001597, A003963, A056239, A064573, A112798, A246547, A246655, A279787, A320322, A320325, A320698, A320700.

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

A320700 Odd numbers whose product of prime indices is a nonprime prime power (A246547).

Original entry on oeis.org

7, 9, 19, 21, 23, 25, 27, 49, 53, 57, 63, 81, 97, 103, 115, 121, 125, 131, 133, 147, 159, 171, 189, 227, 243, 289, 311, 343, 361, 371, 393, 399, 419, 441, 477, 513, 515, 529, 567, 575, 625, 661, 691, 719, 729, 917, 931, 933, 961, 1007, 1009, 1029, 1067, 1083

Offset: 1

Gus Wiseman, Oct 19 2018

nonn

			The sequence of all integer partitions whose Heinz numbers belong to the sequence begins: (4), (2,2), (8), (4,2), (9), (3,3), (2,2,2), (4,4), (16), (8,2), (4,2,2), (2,2,2,2), (25), (27), (9,3), (5,5), (3,3,3), (32), (8,4), (4,4,2), (16,2), (8,2,2), (4,2,2,2), (49), (2,2,2,2,2)

Index entries for sequences computed from indices in prime factorization

Cf. A000720, A000961, A001597, A003963, A056239, A064573, A112798, A246547, A279787, A320322, A320325, A320698, A320699.

Select[Range[1000],With[{x=Times@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]^k]},OddQ[#]&&!PrimeQ[x]&&PrimePowerQ[x]]&]

A322546 Numbers k such that every integer partition of k contains a 1 or a prime power.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 19, 23

Offset: 1

Gus Wiseman, Dec 14 2018

			24 does not belong to the sequence because there are integer partitions of 24 containing no 1's or prime powers, namely: (24), (18,6), (14,10), (12,12), (12,6,6), (6,6,6,6).

Cf. A000607, A002095, A023893, A023894, A064573, A078135, A101417, A246655, A320322, A322452, A322454, A322547.

nn=100;
ser=Product[If[n==1||PrimePowerQ[n],1,1/(1-x^n)],{n,nn}];
Join@@Position[CoefficientList[Series[ser,{x,0,nn}],x],0]-1

A357490 Numbers k such that the k-th composition in standard order has integer geometric mean.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 10, 15, 16, 17, 24, 31, 32, 36, 42, 63, 64, 69, 70, 81, 88, 98, 104, 127, 128, 136, 170, 255, 256, 277, 278, 282, 292, 325, 326, 337, 344, 354, 360, 394, 418, 424, 511, 512, 513, 514, 515, 528, 547, 561, 568, 640, 682, 768, 769, 785, 792, 896

Offset: 1

Gus Wiseman, Oct 16 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their corresponding compositions begin:
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   7: (1,1,1)
   8: (4)
  10: (2,2)
  15: (1,1,1,1)
  16: (5)
  17: (4,1)
  24: (1,4)
  31: (1,1,1,1,1)
  32: (6)
  36: (3,3)
  42: (2,2,2)
  63: (1,1,1,1,1,1)
  64: (7)
  69: (4,2,1)

For regular mean we have A096199, counted by A271654 (partitions A067538).
Subsets whose geometric mean is an integer are counted by A326027.
The unordered version (partitions) is A326623, counted by A067539.
The strict case is counted by A339452, partitions A326625.
These compositions are counted by A357710.
A078175 lists numbers whose prime factors have integer average.
A320322 counts partitions whose product is a perfect power.
Cf. A051293, A078174, A102627, A301987, A326622, A326624, A326028, A326567/A326568, A326641, A326645, A335405, A357184.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],IntegerQ[GeometricMean[stc[#]]]&]

A358331 Number of integer partitions of n with arithmetic and geometric mean differing by one.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 2, 0, 2, 0, 1, 1, 0, 3, 3, 0, 0, 2, 2, 0, 4, 0, 0, 5, 0, 0, 4, 5, 4, 3, 2, 0, 3, 3, 10, 4, 0, 0, 7, 0, 0, 16, 2, 4, 4, 0, 0, 5, 24, 0, 6, 0, 0, 9, 0, 27, 10, 0, 7, 7, 1, 0, 44

Offset: 0

Gus Wiseman, Nov 09 2022

nonn

The arithmetic and geometric mean from such partition is a positive integer. - David A. Corneth, Nov 11 2022

			The a(30) = 2 through a(36) = 3 partitions (C = 12, G = 16):
  (888222)      .  (99333311)  (G2222222111)  .  (C9662)    (G884)
  (8844111111)                                   (C9833)    (888222111111)
                                                 (8884421)  (G42222221111)

Chai Wah Wu, Table of n, a(n) for n = 0..143
Wikipedia, Geometric mean

The version for subsets seems to be close to A178832.
These partitions are ranked by A358332.
A000041 counts partitions.
A067538 counts partitions with integer average, ranked by A316413.
A067539 counts partitions with integer geometric mean, ranked by A326623.
A078175 lists numbers whose prime factors have integer average.
A320322 counts partitions whose product is a perfect power.
Cf. A078174, A102627, A271654, A326027, A326624, A326625, A326028, A326641, A326645, A357710.

Table[Length[Select[IntegerPartitions[n],Mean[#]==1+GeometricMean[#]&]],{n,0,30}]

a(n) = if (n, my(nb=0,vp); forpart(p=n, vp=Vec(p); if (vecsum(vp)/#p == 1 + sqrtn(vecprod(vp), #p), nb++)); nb, 0); \\ Michel Marcus, Nov 11 2022

from math import prod
from sympy import divisors, integer_nthroot
from sympy.utilities.iterables import partitions
def A358331(n):
    divs = {d:n//d-1 for d in divisors(n,generator=True)}
    return sum(1 for s,p in partitions(n,m=max(divs,default=0),size=True) if s in divs and (t:=integer_nthroot(prod(a**b for a, b in p.items()),s))[1] and divs[s]==t[0]) # Chai Wah Wu, Sep 24 2023

a(61)-a(80) from Giorgos Kalogeropoulos, Nov 11 2022

a(81)-a(84) from Chai Wah Wu, Sep 24 2023

cp's OEIS Frontend

A322528 Number of integer partitions of n whose parts all have the same number of prime factors (counted with multiplicity) and whose product of parts is a power of a squarefree number (A072774).

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A322529 Number of integer partitions of n whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.

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A325045 Number of factorizations of n whose conjugate as an integer partition has no ones.

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A322453 Number of factorizations of n into factors > 1 using only primes and perfect powers.

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A339452 Number of compositions (ordered partitions) of n into distinct parts such that the geometric mean of the parts is an integer.

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A320699 Numbers whose product of prime indices is a nonprime prime power (A246547).

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A320700 Odd numbers whose product of prime indices is a nonprime prime power (A246547).

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A322546 Numbers k such that every integer partition of k contains a 1 or a prime power.

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