A352140 Numbers whose prime factorization has all even prime indices and all odd exponents.

1, 3, 7, 13, 19, 21, 27, 29, 37, 39, 43, 53, 57, 61, 71, 79, 87, 89, 91, 101, 107, 111, 113, 129, 131, 133, 139, 151, 159, 163, 173, 181, 183, 189, 193, 199, 203, 213, 223, 229, 237, 239, 243, 247, 251, 259, 263, 267, 271, 273, 281, 293, 301, 303, 311, 317

Offset: 1

Author

Gus Wiseman, Mar 11 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.

Also Heinz numbers of integer partitions with all even parts and all odd multiplicities, counted by A055922 aerated.

All terms are odd. - Michael S. Branicky, Mar 12 2022

Examples

			The terms together with their prime indices begin:
      1 = 1
      3 = prime(2)^1
      7 = prime(4)^1
     13 = prime(6)^1
     19 = prime(8)^1
     21 = prime(4)^1 prime(2)^1
     27 = prime(2)^3
     29 = prime(10)^1
     37 = prime(12)^1
     39 = prime(6)^1 prime(2)^1
     43 = prime(14)^1
     53 = prime(16)^1
     57 = prime(8)^1 prime(2)^1
     61 = prime(18)^1
     71 = prime(20)^1
		

Crossrefs

The restriction to primes is A031215.

These partitions are counted by A055922 (aerated).

The first condition alone is A066207, counted by A035363.

The squarefree case is A258117.

The second condition alone is A268335, counted by A055922.

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 prime exponents, odd A162642.

A257991 counts odd prime 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.

A352141 = even indices with even exponents, counted by A035444.

A352142 = odd indices with odd exponents, counted by A117958.

Cf. A000720, A028260, A055396, A061395, A181819, A195017, A241638, A276078, A324517, A324524, A324525, A325698.

Programs

Mathematica

Select[Range[100],And@@EvenQ/@PrimePi/@First/@FactorInteger[#]&&And@@OddQ/@Last/@FactorInteger[#]&]

Python

from sympy import factorint, primepi
def ok(n):
    if n%2 == 0: return False
    return all(primepi(p)%2==0 and e%2==1 for p, e in factorint(n).items())
print([k for k in range(318) if ok(k)]) # Michael S. Branicky, Mar 12 2022

Formula

Intersection of A066207 and A268335.

A257991(a(n)) = A162641(a(n)) = 0.

A162642(a(n)) = A001221(a(n)).

A257992(a(n)) = A001222(a(n)).