A320322 -id:A320322 - cp's OEIS frontend

A321452 Number of integer partitions of n that can be partitioned into two or more blocks with equal sums.

0, 0, 1, 1, 3, 1, 7, 1, 14, 10, 26, 1, 55, 1, 90, 68, 167, 1, 292, 1, 482, 345, 761, 1, 1291, 266, 1949, 1518, 3091, 1, 4793, 1, 7177, 5612, 10566, 2623, 16007, 1, 22912, 18992, 33619, 1, 48529, 1, 68758, 59187, 96571, 1, 137489, 11418, 189979, 167502, 264299

Offset: 0

Gus Wiseman, Nov 10 2018

nonn

a(n) = 1 if and only if n is prime. - Chai Wah Wu, Nov 12 2018

			The a(2) = 1 through a(9) = 10 partitions:
  (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)        (333)
               (211)            (222)                (422)       (3321)
               (1111)           (321)                (431)       (32211)
                                (2211)               (2222)      (33111)
                                (3111)               (3221)      (222111)
                                (21111)              (3311)      (321111)
                                (111111)             (4211)      (2211111)
                                                     (22211)     (3111111)
                                                     (32111)     (21111111)
                                                     (41111)     (111111111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)
The partition (32111) can be partitioned as ((13)(112)), and the blocks both sum to 4, so (32111) is counted under a(8).

Cf. A000041, A265947, A276024, A279787, A305551, A306017, A317141, A320322, A321451, A321453, A321454, A321455.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[IntegerPartitions[n],Length[Select[facs[Times@@Prime/@#],SameQ@@hwt/@#&]]>1&]],{n,10}]

a(n) = A000041(n) - A321451(n).

a(26)-a(52) from Alois P. Heinz, Nov 11 2018

A067539 Number of partitions of n in which, if the number of parts is k, the product of the parts is the k-th power of some positive integer.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 3, 4, 4, 8, 3, 8, 5, 7, 8, 8, 7, 9, 8, 17, 11, 11, 8, 16, 17, 17, 14, 18, 17, 26, 19, 24, 20, 30, 28, 32, 27, 37, 35, 48, 37, 45, 37, 51, 51, 58, 50, 64, 62, 83, 73, 84, 69, 91, 89, 101, 97, 116, 111, 136, 123, 142, 138, 160, 161, 181, 171, 205, 199, 231, 221

Offset: 1

Naohiro Nomoto, Jan 27 2002

a(n) is the number of integer partitions of n whose geometric mean is an integer. - Gus Wiseman, Jul 19 2019

			From _Gus Wiseman_, Jul 19 2019: (Start)
The a(1) = 1 through a(8) = 4 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (41)     (33)      (421)      (44)
                    (1111)  (11111)  (222)     (1111111)  (2222)
                                     (111111)             (11111111)
(End)

Wikipedia, Geometric mean

Partitions with integer average are A067538.
Subsets whose geometric mean is an integer are A326027.
The Heinz numbers of these partitions are A326623.
The strict case is A326625.
Cf. A000041, A102627, A320322, A326028, A326641.

Table[Length[Select[IntegerPartitions[n],IntegerQ[GeometricMean[#]]&]],{n,30}] (* Gus Wiseman, Jul 19 2019 *)

from math import prod
from sympy import integer_nthroot
from sympy.utilities.iterables import partitions
def A067539(n): return sum(1 for s,p in partitions(n,size=True) if integer_nthroot(prod(a**b for a, b in p.items()),s)[1]) # Chai Wah Wu, Sep 24 2023

Terms a(61) onwards from Max Alekseyev, Feb 06 2010

A064573 Number of partitions of n into parts which are all powers of the same prime.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 13, 15, 20, 21, 29, 30, 37, 40, 50, 51, 64, 65, 80, 84, 99, 100, 123, 125, 146, 151, 178, 179, 212, 213, 249, 255, 292, 295, 348, 349, 396, 404, 466, 467, 535, 536, 611, 622, 697, 698, 801, 803, 900, 910, 1025, 1026, 1152, 1156, 1298, 1311

Offset: 1

Marc LeBrun, Sep 20 2001

The exponents cannot all be zero.

			a(5)=5: 5^1, 3^1+2*3^0, 2^2+1, 2*2^1+1, 2^1+3*2^0
From _Gus Wiseman_, Oct 10 2018: (Start)
The a(2) = 1 through a(9) = 15 integer partitions:
  (2)  (3)   (4)    (5)     (33)     (7)       (8)        (9)
       (21)  (22)   (41)    (42)     (331)     (44)       (81)
             (31)   (221)   (51)     (421)     (71)       (333)
             (211)  (311)   (222)    (511)     (422)      (441)
                    (2111)  (411)    (2221)    (2222)     (711)
                            (2211)   (4111)    (3311)     (4221)
                            (3111)   (22111)   (4211)     (22221)
                            (21111)  (31111)   (5111)     (33111)
                                     (211111)  (22211)    (42111)
                                               (41111)    (51111)
                                               (221111)   (222111)
                                               (311111)   (411111)
                                               (2111111)  (2211111)
                                                          (3111111)
                                                          (21111111)
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A028422, A064572, A064574, A064575, A064576, A064577, A319071, A320322, A320325.

Table[Length[Select[IntegerPartitions[n],PrimePowerQ[Times@@#]&]],{n,30}] (* Gus Wiseman, Oct 10 2018 *)

first(n)={Vec(sum(k=2, n, if(isprime(k), 1/prod(r=0, logint(n,k), 1-x^(k^r) + O(x*x^n)) - 1/(1-x), 0)), -n)} \\ Andrew Howroyd, Dec 29 2017

G.f.: Sum_{k>=1} 1/(Product_{r>=0} 1-x^(prime(k)^r)) - 1/(1-x). - Andrew Howroyd, Dec 29 2017

Name clarified by Andrew Howroyd, Dec 29 2017

A321451 Number of integer partitions of n that cannot be partitioned into two or more blocks with equal sums.

Original entry on oeis.org

1, 1, 1, 2, 2, 6, 4, 14, 8, 20, 16, 55, 22, 100, 45, 108, 64, 296, 93, 489, 145, 447, 241, 1254, 284, 1692, 487, 1492, 627, 4564, 811, 6841, 1172, 4531, 1744, 12260, 1970, 21636, 3103, 12193, 3719, 44582, 4645, 63260, 6417, 29947, 8987, 124753, 9784, 162107, 14247

Offset: 0

Gus Wiseman, Nov 10 2018

nonn

			The a(1) = 1 through a(9) = 20 partitions:
  (1)  (2)  (3)   (4)   (5)     (6)    (7)       (8)     (9)
            (21)  (31)  (32)    (42)   (43)      (53)    (54)
                        (41)    (51)   (52)      (62)    (63)
                        (221)   (411)  (61)      (71)    (72)
                        (311)          (322)     (332)   (81)
                        (2111)         (331)     (521)   (432)
                                       (421)     (611)   (441)
                                       (511)     (5111)  (522)
                                       (2221)            (531)
                                       (3211)            (621)
                                       (4111)            (711)
                                       (22111)           (3222)
                                       (31111)           (4221)
                                       (211111)          (4311)
                                                         (5211)
                                                         (6111)
                                                         (22221)
                                                         (42111)
                                                         (51111)
                                                         (411111)
A complete list of all multiset partitions of the partition (2111) into two or more blocks is: ((1)(112)), ((2)(111)), ((11)(12)), ((1)(1)(12)), ((1)(2)(11)), ((1)(1)(1)(2)). None of these has equal block-sums, so (2111) is counted toward a(5).
On the other hand, the partition (321) can be partitioned as ((12)(3)), which has two or more blocks and equal block-sums, so (321) is not counted toward a(6).

Cf. A000041, A265947, A276024, A279787, A305551, A306017, A317141, A320322, A321452, A321453, A321454, A321455.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[IntegerPartitions[n],Length[Select[facs[Times@@Prime/@#],SameQ@@hwt/@#&]]==1&]],{n,10}]

a(n) = A000041(n) - A321452(n).

a(33)-a(50) from Alois P. Heinz, Nov 11 2018

A321454 Numbers that can be factored into two or more factors all having the same sum of prime indices.

Original entry on oeis.org

4, 8, 9, 12, 16, 25, 27, 30, 32, 36, 40, 48, 49, 63, 64, 70, 81, 84, 90, 100, 108, 112, 120, 121, 125, 128, 144, 150, 154, 160, 165, 169, 180, 192, 196, 198, 200, 210, 216, 220, 225, 240, 243, 252, 256, 264, 270, 273, 280, 286, 288, 289, 300, 320, 324, 325

Offset: 1

Gus Wiseman, Nov 10 2018

nonn

Also Heinz numbers of integer partitions that can be partitioned into two or more blocks with equal sums. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

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 sum of prime indices of n is A056239(n).

			The sequence of all integer partitions that can be partitioned into two or more blocks with equal sums begins: (11), (111), (22), (211), (1111), (33), (222), (321), (11111), (2211), (3111), (21111), (44), (422), (111111), (431), (2222), (4211), (3221), (3311), (22211), (41111), (32111), (55), (333), (1111111), (221111), (3321), (541), (311111), (532), (66), (32211), (2111111), (4411), (5221), (33111).
The Heinz number of (32111) is 120, which has factorization (10*12) corresponding to the multiset partition ((13)(112)) whose blocks have equal sums, so 120 belongs to the sequence.

Positions of terms > 1 in A321455.

Cf. A056239, A112798, A276024, A279787, A305551, A306017, A317144, A320322, A321451, A321452, A321453.

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

A321453 Numbers that cannot be factored into two or more factors all having the same sum of prime indices.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83, 85

Offset: 1

Gus Wiseman, Nov 10 2018

nonn

Also Heinz numbers of integer partitions that cannot be partitioned into two or more blocks with equal sums. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

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 sum of prime indices of n is A056239(n).

			The sequence of all integer partitions that cannot be partitioned into two or more blocks with equal sums begins: (1), (2), (3), (21), (4), (31), (5), (6), (41), (32), (7), (221), (8), (311), (42), (51), (9), (2111), (61), (411).

Positions of 1's in A321455.

Cf. A056239, A112798, A276024, A279787, A305551, A306017, A317144, A320322, A321451, A321452, A321454.

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

A320325 Numbers whose product of prime indices is a perfect power.

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, 151, 152, 159, 161, 162, 168, 169, 171, 175, 183, 184, 185, 189, 194, 195, 196

Offset: 1

Gus Wiseman, Oct 10 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.

			The terms together with their corresponding multiset multisystems (A302242):
   7: {{1,1}}
   9: {{1},{1}}
  14: {{},{1,1}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  28: {{},{},{1,1}}
  36: {{},{},{1},{1}}
  38: {{},{1,1,1}}
  42: {{},{1},{1,1}}
  46: {{},{2,2}}
  49: {{1,1},{1,1}}
  50: {{},{2},{2}}
  53: {{1,1,1,1}}
  54: {{},{1},{1},{1}}
  56: {{},{},{},{1,1}}
  57: {{1},{1,1,1}}
  63: {{1},{1},{1,1}}
  72: {{},{},{},{1},{1}}
  76: {{},{},{1,1,1}}
  81: {{1},{1},{1},{1}}

Cf. A000720, A001222, A001597, A003963, A056239, A064573, A112798, A302242, A305551, A306017, A319056, A319066, A320322, A320323, A320324.

Select[Range[100],GCD@@FactorInteger[Times@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]^k]][[All,2]]>1&]

A319169 Number of integer partitions of n whose parts all have the same number of prime factors, counted with multiplicity.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 7, 11, 11, 14, 15, 20, 19, 26, 27, 34, 35, 43, 45, 59, 60, 72, 77, 94, 98, 118, 125, 148, 158, 184, 198, 233, 245, 282, 308, 353, 374, 428, 464, 525, 566, 635, 686, 779, 832, 930, 1005, 1123, 1208, 1345, 1451, 1609, 1732, 1912

Offset: 0

Gus Wiseman, Oct 10 2018

nonn

			The a(1) = 1 through a(9) = 6 integer partitions:
  1  2   3    4     5      6       7        8         9
     11  111  22    32     33      52       44        72
              1111  11111  222     322      53        333
                           111111  1111111  332       522
                                            2222      3222
                                            11111111  111111111

Alois P. Heinz, Table of n, a(n) for n = 0..2500 (first 101 terms from Chai Wah Wu)

Cf. A000607, A001222, A003963, A064573, A279787, A305551, A319056, A319066, A319071, A320322, A320324.

b:= proc(n, i, f) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, f)+(o-> `if`(f in {0, o}, b(n-i, min(i, n-i),
     `if`(f=0, o, f)), 0))(numtheory[bigomega](i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..75);  # Alois P. Heinz, Dec 15 2018

Table[Length[Select[IntegerPartitions[n],SameQ@@PrimeOmega/@#&]],{n,30}]
(* Second program: *)
b[n_, i_, f_] := b[n, i, f] = If[n == 0, 1, If[i < 1, 0,
     b[n, i-1, f] + Function[o, If[f == 0 || f == o, b[n-i, Min[i, n-i],
     If[f == 0, o, f]], 0]][PrimeOmega[i]]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 75] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

a(51)-a(58) from Chai Wah Wu, Nov 12 2018

A078135 Numbers which cannot be written as a sum of squares > 1.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 19, 23

Offset: 1

Reinhard Zumkeller, Nov 19 2002

Numbers such that A078134(n)=0.
"Numbers which cannot be written as sum of squares > 1" is equivalent to "Numbers which cannot be written as sum of squares of primes." Equivalently, numbers which can be written as the sum of nonzero squares can also be written as sum of the squares of primes." cf. A090677 = number of ways to partition n into sums of squares of primes. - Jonathan Vos Post, Sep 20 2006
The sequence is finite with a(12)=23 as last member. Proof: When k=a^2+b^2+..., k+4 = 2^2+a^2+b^2+... If k can be written as sum of the squares of primes, k+4 also has this property. As 24,25,26,27 have the property, by induction, all numbers > 23 can be written as sum of squares>1. - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Apr 07 2007
Also, numbers which cannot be written as sum of squares of 2 and 3 (see A078137 for the proof). Explicit representation as sum of squares of primes, or rather of squares of 2 and 3, for numbers m>23: we have m=c*2^2+d*3^2, where c:=(floor(m/4) - 2*(m mod 4))>=0, d:=m mod 4. For that, the finiteness of the sequence is proved constructively. - Hieronymus Fischer, Nov 11 2007
Also numbers n such that every integer partition of n contains a squarefree number. For example, 21 does not belong to the sequence because there are integer partitions of 21 containing no squarefree numbers, namely: (12,9), (9,8,4), (9,4,4,4). - Gus Wiseman, Dec 14 2018

Eric Weisstein's World of Mathematics, Square Number.
Index entries for sequences related to sums of squares

Cf. A000290, A001422, A078137, A078139, A078136, A078129.
Cf. A090677.
Cf. A078134, A078139, A090677, A078137, A134754, A134755.
Cf. A005117, A101417, A320322, A322526, A322546, A322547.

nn=100;
ser=Product[If[SquareFreeQ[n],1,1/(1-x^n)],{n,nn}];
Join@@Position[CoefficientList[Series[ser,{x,0,nn}],x],0]-1 (* Gus Wiseman, Dec 14 2018 *)

A090677(a(n)) = 0. - Jonathan Vos Post, Sep 20 2006 [corrected by Joerg Arndt, Dec 16 2018]

A033183(a(n)) = 0. - Reinhard Zumkeller, Nov 07 2009

A325039 Number of integer partitions of n with the same product of parts as their conjugate.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 6, 2, 2, 4, 3, 5, 7, 6, 5, 7, 9, 10, 11, 18, 16, 19, 19, 16, 20, 20, 28, 39, 28, 40, 53, 45, 52, 59, 71, 61, 73, 97, 102, 95, 112, 131, 137, 148, 140, 166, 199, 181, 238, 251, 255, 289, 339, 344, 381, 398, 422, 464, 541, 555, 628, 677, 732

Offset: 0

Gus Wiseman, Mar 25 2019

nonn

For example, the partition (6,4,1) with product 24 has conjugate (3,2,2,2,1,1) with product also 24.

The Heinz numbers of these partitions are given by A325040.

			The a(8) = 6 through a(15) = 6 integer partitions:
  (44)    (333)    (4321)   (641)     (4422)    (4432)     (6431)
  (332)   (51111)  (52111)  (4331)    (53211)   (6421)     (8411)
  (431)                     (322211)  (621111)  (53311)    (54221)
  (2222)                    (611111)            (432211)   (433211)
  (3221)                                        (7111111)  (632111)
  (4211)                                                   (7211111)
                                                           (42221111)

Cf. A001055, A064573, A122111, A296150, A318950, A319000, A320322, A321649.

Cf. A325040, A325041, A325042, A325045.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],Times@@#==Times@@conj[#]&]],{n,0,30}]

More terms from Jinyuan Wang, Jun 27 2020

cp's OEIS Frontend

A321452 Number of integer partitions of n that can be partitioned into two or more blocks with equal sums.

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A067539 Number of partitions of n in which, if the number of parts is k, the product of the parts is the k-th power of some positive integer.

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A064573 Number of partitions of n into parts which are all powers of the same prime.

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A321451 Number of integer partitions of n that cannot be partitioned into two or more blocks with equal sums.

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A321454 Numbers that can be factored into two or more factors all having the same sum of prime indices.

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A321453 Numbers that cannot be factored into two or more factors all having the same sum of prime indices.

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A320325 Numbers whose product of prime indices is a perfect power.

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A319169 Number of integer partitions of n whose parts all have the same number of prime factors, counted with multiplicity.

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