A352141 Numbers whose prime factorization has all even indices and all even exponents.

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

Offset: 1

Author

Gus Wiseman, Mar 18 2022

Keywords

• nonn

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.

Examples

			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
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

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.

Cf. A000720, A028260, A055396, A061395, A181819, A195017, A241638, A268335, A276078, A324524, A324525, A324588, A325698, A325700, A352143.

Programs

Mathematica

Select[Range[1000],#==1||And@@EvenQ/@PrimePi/@First/@FactorInteger[#]&&And@@EvenQ/@Last/@FactorInteger[#]&]

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

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