A325132 -id:A325132 - cp's OEIS frontend

A114640 Number of partitions of n such that the set of parts and the set of multiplicities of parts are equal.

1, 1, 0, 0, 2, 1, 1, 0, 1, 1, 3, 2, 3, 3, 5, 0, 4, 5, 2, 3, 8, 6, 5, 10, 9, 9, 16, 14, 12, 16, 17, 10, 17, 15, 16, 19, 35, 17, 34, 37, 40, 31, 54, 36, 60, 61, 58, 63, 88, 58, 88, 87, 91, 84, 115, 93, 116, 108, 115, 130, 190, 143, 165, 214, 219, 200, 255, 240

Offset: 0

Vladeta Jovovic, Feb 18 2006

nonn

The Heinz numbers of these partitions are given by A109297. - Gus Wiseman, Apr 02 2019

			From _Gus Wiseman_, Apr 02 2019: (Start)
The initial terms count the following integer partitions:
   0: ()
   1: (1)
   4: (22)
   4: (211)
   5: (221)
   6: (3111)
   8: (41111)
   9: (333)
  10: (511111)
  10: (3331)
  10: (322111)
  11: (332111)
  11: (322211)
  12: (6111111)
  12: (4221111)
  12: (33222)
  13: (33322)
  13: (333211)
  13: (332221)
  14: (71111111)
  14: (52211111)
  14: (4421111)
  14: (4222211)
  14: (333221)
(End)

Cf. A052335, A087153, A109297, A114639, A115584, A117144, A276429, A324572, A325132, A336031.

Table[Length[Select[IntegerPartitions[n],Union[#]==Union[Length/@Split[#]]&]],{n,0,30}] (* Gus Wiseman, Apr 02 2019 *)

More terms from Alois P. Heinz, Aug 09 2016

A054744 p-full numbers: numbers such that if any prime p divides it, then so does p^p.

Original entry on oeis.org

1, 4, 8, 16, 27, 32, 64, 81, 108, 128, 216, 243, 256, 324, 432, 512, 648, 729, 864, 972, 1024, 1296, 1728, 1944, 2048, 2187, 2592, 2916, 3125, 3456, 3888, 4096, 5184, 5832, 6561, 6912, 7776, 8192, 8748, 10368, 11664, 12500, 13824, 15552, 15625, 16384

Offset: 1

James Sellers, Apr 22 2000

A027748(a(n),k) <= A124010(a(n),k), 1<=k<=A001221(a(n)). [Reinhard Zumkeller, Apr 28 2012]

Heinz numbers of integer partitions where the multiplicity of each part k is at least prime(k). These partitions are counted by A325132. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Apr 02 2019

			8 is an element because 8 = 2^3 and 2<=3, while 25 is not an element because 25 = 5^2 and 5>2.
From _Gus Wiseman_, Apr 02 2019: (Start)
The sequence of terms together with their prime indices begins:
    1: {}
    4: {1,1}
    8: {1,1,1}
   16: {1,1,1,1}
   27: {2,2,2}
   32: {1,1,1,1,1}
   64: {1,1,1,1,1,1}
   81: {2,2,2,2}
  108: {1,1,2,2,2}
  128: {1,1,1,1,1,1,1}
  216: {1,1,1,2,2,2}
  243: {2,2,2,2,2}
  256: {1,1,1,1,1,1,1,1}
  324: {1,1,2,2,2,2}
  432: {1,1,1,1,2,2,2}
  512: {1,1,1,1,1,1,1,1,1}
  648: {1,1,1,2,2,2,2}
  729: {2,2,2,2,2,2}
  864: {1,1,1,1,1,2,2,2}
  972: {1,1,2,2,2,2,2}
(End)

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A048103, A001694, A036966.
Cf. A100716.
Cf. A056239, A109298, A112798, A115584, A117144, A124010, A276078, A324525, A324571, A325127, A325128, A325130, A325131, A325132.

a054744 n = a054744_list !! (n-1)
a054744_list = filter (\x -> and $
   zipWith (<=) (a027748_row x) (map toInteger $ a124010_row x)) [1..]
-- Reinhard Zumkeller, Apr 28 2012

Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k>=p]&] (* Gus Wiseman, Apr 02 2019 *)

If n = Product p_i^e_i then p_i<=e_i for all i.

Sum_{n>=1} 1/a(n) = Product_{p prime} 1 + 1/(p^(p-1)*(p-1)) = 1.58396891058853238595.... - Amiram Eldar, Oct 24 2020

cp's OEIS Frontend

A114640 Number of partitions of n such that the set of parts and the set of multiplicities of parts are equal.

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