

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
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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

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], #==1And@@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)))


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.
A352140 = even indices with odd exponents, counted by A055922 aerated.
Cf. A000720, A028260, A055396, A061395, A181819, A195017, A241638, A268335, A276078, A324524, A324525, A324588, A325698, A325700, A352143.


KEYWORD

nonn


AUTHOR



STATUS

approved



