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A340784
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Heinz numbers of even-length integer partitions of even numbers.
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16
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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EXAMPLE
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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)
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], EvenQ[PrimeOmega[#]]&&EvenQ[Total[primeMS[#]]]&]
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PROG
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(PARI)
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
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CROSSREFS
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Note: A-numbers of Heinz-number sequences are in parentheses below.
The case of prime powers is A056798.
These partitions are counted by A236913.
A061395 selects the maximum prime index.
A072233 counts partitions by sum and length.
A112798 lists the prime indices of each positive integer.
- Even -
A034008 counts compositions of even length.
A339846 counts factorizations of even length.
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.
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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