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.
Also Heinz numbers of integer partitions with all odd parts and all even multiplicities, counted by A035457 (see Emeric Deutsch's comment there).
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
EXAMPLE
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
MATHEMATICA
Select[Range[1000], #==1||And@@OddQ/@PrimePi/@First/@FactorInteger[#]&&And@@EvenQ/@Last/@FactorInteger[#]&]
PROG
(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())
print([k for k in range(15500) if ok(k)]) # Michael S. Branicky, Mar 12 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 11 2022
STATUS
approved