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
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
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)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 18 2022
STATUS
approved