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A352141 Numbers whose prime factorization has all even indices and all even exponents. 7
1, 9, 49, 81, 169, 361, 441, 729, 841, 1369, 1521, 1849, 2401, 2809, 3249, 3721, 3969, 5041, 6241, 6561, 7569, 7921, 8281, 10201, 11449, 12321, 12769, 13689, 16641, 17161, 17689, 19321, 21609, 22801, 25281, 26569, 28561, 29241, 29929, 32761, 33489, 35721 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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 partitions with all even parts and all even multiplicities, counted by A035444.
LINKS
FORMULA
Intersection of A000290 and A066207.
A257991(a(n)) = A162642(a(n)) = 0.
A257992(a(n)) = A001222(a(n)).
A162641(a(n)) = A001221(a(n)).
Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2*k)^2) = 1.163719... . - Amiram Eldar, Sep 19 2022
EXAMPLE
The terms together with their prime indices begin:
1 = 1
9 = prime(2)^2
49 = prime(4)^2
81 = prime(2)^4
169 = prime(6)^2
361 = prime(8)^2
441 = prime(2)^2 prime(4)^2
729 = prime(2)^6
841 = prime(10)^2
1369 = prime(12)^2
1521 = prime(2)^2 prime(6)^2
1849 = prime(14)^2
2401 = prime(4)^4
2809 = prime(16)^2
3249 = prime(2)^2 prime(8)^2
3721 = prime(18)^2
3969 = prime(2)^4 prime(4)^2
MATHEMATICA
Select[Range[1000], #==1||And@@EvenQ/@PrimePi/@First/@FactorInteger[#]&&And@@EvenQ/@Last/@FactorInteger[#]&]
PROG
(Python)
from itertools import count, islice
from sympy import factorint, primepi
def A352141_gen(startvalue=1): # generator of terms >= startvalue
return filter(lambda k:all(map(lambda x: not (x[1]%2 or primepi(x[0])%2), factorint(k).items())), count(max(startvalue, 1)))
A352141_list = list(islice(A352141_gen(), 30)) # Chai Wah Wu, Mar 18 2022
CROSSREFS
The second condition alone (all even exponents) is A000290, counted by A035363.
The restriction to primes is A031215.
These partitions are counted by A035444.
The first condition alone is A066207, counted by A035363, squarefree A258117.
A056166 = exponents all prime, counted by A055923.
A066208 = prime indices all odd, counted by A000009.
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162641 counts even exponents, odd A162642.
A257991 counts odd indices, even A257992.
A325131 = disjoint indices from exponents, counted by A114639.
A346068 = indices and exponents all prime, counted by A351982.
A351979 = odd indices with even exponents, counted by A035457.
A352140 = even indices with odd exponents, counted by A055922 aerated.
A352142 = odd indices with odd exponents, counted by A117958.
Sequence in context: A028375 A167744 A032598 * A110873 A167716 A087352
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 18 2022
STATUS
approved

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Last modified May 30 17:31 EDT 2024. Contains 372971 sequences. (Running on oeis4.)