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A351979
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Numbers whose prime factorization has all odd prime indices and all even prime exponents.
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5
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1, 4, 16, 25, 64, 100, 121, 256, 289, 400, 484, 529, 625, 961, 1024, 1156, 1600, 1681, 1936, 2116, 2209, 2500, 3025, 3481, 3844, 4096, 4489, 4624, 5329, 6400, 6724, 6889, 7225, 7744, 8464, 8836, 9409, 10000, 10609, 11881, 12100, 13225, 13924, 14641, 15376
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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 odd parts and all even multiplicities, counted by A035457 (see Emeric Deutsch's comment there).
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LINKS
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FORMULA
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Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2*k-1)^2) = 1.4135142... . - Amiram Eldar, Sep 19 2022
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EXAMPLE
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The terms together with their prime indices begin:
1: 1
4: prime(1)^2
16: prime(1)^4
25: prime(3)^2
64: prime(1)^6
100: prime(1)^2 prime(3)^2
121: prime(5)^2
256: prime(1)^8
289: prime(7)^2
400: prime(1)^4 prime(3)^2
484: prime(1)^2 prime(5)^2
529: prime(9)^2
625: prime(3)^4
961: prime(11)^2
1024: prime(1)^10
1156: prime(1)^2 prime(7)^2
1600: prime(1)^6 prime(3)^2
1681: prime(13)^2
1936: prime(1)^4 prime(5)^2
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MATHEMATICA
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Select[Range[1000], #==1||And@@OddQ/@PrimePi/@First/@FactorInteger[#]&&And@@EvenQ/@Last/@FactorInteger[#]&]
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PROG
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(Python)
from sympy import factorint, primepi
def ok(n):
return all(primepi(p)%2==1 and e%2==0 for p, e in factorint(n).items())
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CROSSREFS
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The second condition alone (exponents all even) is A000290, counted by A035363.
The distinct prime factors of terms all come from A031368.
The first condition alone (indices all odd) is A066208, counted by A000009.
A352140 = even indices with odd exponents, counted by A055922 (aerated).
Cf. A000720, A028260, A045931, A055396, A061395, A106529, A181819, A195017, A276078, A324588, 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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