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A352142
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Numbers whose prime factorization has all odd indices and all odd exponents.
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8
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1, 2, 5, 8, 10, 11, 17, 22, 23, 31, 32, 34, 40, 41, 46, 47, 55, 59, 62, 67, 73, 82, 83, 85, 88, 94, 97, 103, 109, 110, 115, 118, 125, 127, 128, 134, 136, 137, 146, 149, 155, 157, 160, 166, 167, 170, 179, 184, 187, 191, 194, 197, 205, 206, 211, 218, 227, 230
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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.
These are the Heinz numbers of integer partitions with all odd parts and all odd multiplicities, counted by A117958.
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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
2 = prime(1)
5 = prime(3)
8 = prime(1)^3
10 = prime(1) prime(3)
11 = prime(5)
17 = prime(7)
22 = prime(1) prime(5)
23 = prime(9)
31 = prime(11)
32 = prime(1)^5
34 = prime(1) prime(7)
40 = prime(1)^3 prime(3)
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MATHEMATICA
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Select[Range[100], #==1||And@@OddQ/@PrimePi/@First/@FactorInteger[#]&&And@@OddQ/@Last/@FactorInteger[#]&]
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PROG
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(Python)
from itertools import count, islice
from sympy import primepi, factorint
def A352142_gen(startvalue=1): # generator of terms >= startvalue
return filter(lambda k:all(map(lambda x:x[1]%2 and primepi(x[0])%2, factorint(k).items())), count(max(startvalue, 1)))
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CROSSREFS
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The restriction to primes is A031368.
These partitions are counted by A117958.
A352140 = even indices with odd exponents, counted by A055922 aerated.
A352143 = odd indices with odd conjugate indices, counted by A053253 aerated.
Cf. A000720, A028260, A055396, A061395, A106529, A181819, A195017, A241638, A276078, A324517, A324524, A324525, A325698, A325700.
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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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