A340784 Heinz numbers of even-length integer partitions of even numbers.

1, 4, 9, 10, 16, 21, 22, 25, 34, 36, 39, 40, 46, 49, 55, 57, 62, 64, 81, 82, 84, 85, 87, 88, 90, 91, 94, 100, 111, 115, 118, 121, 129, 133, 134, 136, 144, 146, 155, 156, 159, 160, 166, 169, 183, 184, 187, 189, 194, 196, 198, 203, 205, 206, 210, 213, 218, 220

Offset: 1

Author

Gus Wiseman, Jan 30 2021

Keywords

• nonn

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are positive integers whose number of prime indices and sum of prime indices are both even, counting multiplicity in both cases.

A multiplicative semigroup: if m and n are in the sequence, then so is m*n. - Antti Karttunen, Jul 28 2024

Examples

			The sequence of partitions together with their Heinz numbers begins:
      1: ()            57: (8,2)            118: (17,1)
      4: (1,1)         62: (11,1)           121: (5,5)
      9: (2,2)         64: (1,1,1,1,1,1)    129: (14,2)
     10: (3,1)         81: (2,2,2,2)        133: (8,4)
     16: (1,1,1,1)     82: (13,1)           134: (19,1)
     21: (4,2)         84: (4,2,1,1)        136: (7,1,1,1)
     22: (5,1)         85: (7,3)            144: (2,2,1,1,1,1)
     25: (3,3)         87: (10,2)           146: (21,1)
     34: (7,1)         88: (5,1,1,1)        155: (11,3)
     36: (2,2,1,1)     90: (3,2,2,1)        156: (6,2,1,1)
     39: (6,2)         91: (6,4)            159: (16,2)
     40: (3,1,1,1)     94: (15,1)           160: (3,1,1,1,1,1)
     46: (9,1)        100: (3,3,1,1)        166: (23,1)
     49: (4,4)        111: (12,2)           169: (6,6)
     55: (5,3)        115: (9,3)            183: (18,2)
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Note: A-numbers of Heinz-number sequences are in parentheses below.

The case of prime powers is A056798.

These partitions are counted by A236913.

The odd version is A160786 (A340931).

A000009 counts partitions into odd parts (A066208).

A001222 counts prime factors.

A047993 counts balanced partitions (A106529).

A056239 adds up prime indices.

A058695 counts partitions of odd numbers (A300063).

A061395 selects the maximum prime index.

A072233 counts partitions by sum and length.

A112798 lists the prime indices of each positive integer.

- Even -

A027187 counts partitions of even length/maximum (A028260/A244990).

A034008 counts compositions of even length.

A035363 counts partitions into even parts (A066207).

A058696 counts partitions of even numbers (A300061).

A067661 counts strict partitions of even length (A030229).

A339846 counts factorizations of even length.

A340601 counts partitions of even rank (A340602).

A340785 counts factorizations into even factors.

A340786 counts even-length factorizations into even factors.

Cf. A026424, A257541, A300272, A326837, A326845, A340385 (A340386), A340604, A353331 (characteristic function), A353332, A353333, A353334.

Squares (A000290) is a subsequence.

Not a subsequence of A329609 (30 is the first term of A329609 not occurring here, and 210 is the first term here not present in A329609).

Positions of even terms in A373381.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],EvenQ[PrimeOmega[#]]&&EvenQ[Total[primeMS[#]]]&]

PARI

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
A353331(n) = ((!(bigomega(n)%2)) && (!(A056239(n)%2)));
isA340784(n) = A353331(n); \\ Antti Karttunen, Apr 14 2022

Formula

Intersection of A028260 and A300061.