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 integer partitions with all odd parts and all odd multiplicities, counted by A117958.
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000
FORMULA
EXAMPLE
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)
MATHEMATICA
Select[Range[100], #==1||And@@OddQ/@PrimePi/@First/@FactorInteger[#]&&And@@OddQ/@Last/@FactorInteger[#]&]
PROG
(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)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 18 2022
STATUS
approved