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A352140
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Numbers whose prime factorization has all even prime indices and all odd exponents.
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6
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1, 3, 7, 13, 19, 21, 27, 29, 37, 39, 43, 53, 57, 61, 71, 79, 87, 89, 91, 101, 107, 111, 113, 129, 131, 133, 139, 151, 159, 163, 173, 181, 183, 189, 193, 199, 203, 213, 223, 229, 237, 239, 243, 247, 251, 259, 263, 267, 271, 273, 281, 293, 301, 303, 311, 317
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OFFSET
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1,2
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COMMENTS
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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, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
Also Heinz numbers of integer partitions with all even parts and all odd multiplicities, counted by A055922 aerated.
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LINKS
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FORMULA
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EXAMPLE
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The terms together with their prime indices begin:
1 = 1
3 = prime(2)^1
7 = prime(4)^1
13 = prime(6)^1
19 = prime(8)^1
21 = prime(4)^1 prime(2)^1
27 = prime(2)^3
29 = prime(10)^1
37 = prime(12)^1
39 = prime(6)^1 prime(2)^1
43 = prime(14)^1
53 = prime(16)^1
57 = prime(8)^1 prime(2)^1
61 = prime(18)^1
71 = prime(20)^1
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MATHEMATICA
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Select[Range[100], And@@EvenQ/@PrimePi/@First/@FactorInteger[#]&&And@@OddQ/@Last/@FactorInteger[#]&]
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PROG
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(Python)
from sympy import factorint, primepi
def ok(n):
if n%2 == 0: return False
return all(primepi(p)%2==0 and e%2==1 for p, e in factorint(n).items())
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CROSSREFS
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The restriction to primes is A031215.
These partitions are counted by A055922 (aerated).
Cf. A000720, A028260, A055396, A061395, A181819, A195017, A241638, A276078, A324517, A324524, A324525, A325698.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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