A352142 Numbers whose prime factorization has all odd indices and all odd exponents.

1, 2, 5, 8, 10, 11, 17, 22, 23, 31, 32, 34, 40, 41, 46, 47, 55, 59, 62, 67, 73, 82, 83, 85, 88, 94, 97, 103, 109, 110, 115, 118, 125, 127, 128, 134, 136, 137, 146, 149, 155, 157, 160, 166, 167, 170, 179, 184, 187, 191, 194, 197, 205, 206, 211, 218, 227, 230

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 integer partitions with all odd parts and all odd multiplicities, counted by A117958.

Examples

			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)
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000

Crossrefs

The restriction to primes is A031368.

The first condition alone is A066208, counted by A000009.

These partitions are counted by A117958.

The squarefree case is A258116, even A258117.

The second condition alone is A268335, counted by A055922.

The even-even version is A352141 counted by A035444.

A000290 = exponents all even, counted by A035363.

A056166 = exponents all prime, counted by A055923.

A066207 = indices all even, counted by A035363 (complement A086543).

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.

A352140 = even indices with odd exponents, counted by A055922 aerated.

A352143 = odd indices with odd conjugate indices, counted by A053253 aerated.

Cf. A000720, A028260, A055396, A061395, A106529, A181819, A195017, A241638, A276078, A324517, A324524, A324525, A325698, A325700.

Programs

Mathematica

Select[Range[100],#==1||And@@OddQ/@PrimePi/@First/@FactorInteger[#]&&And@@OddQ/@Last/@FactorInteger[#]&]

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)))
A352142_list = list(islice(A352142_gen(),30)) # Chai Wah Wu, Mar 18 2022

Formula

Intersection of A066208 and A268335.

A257991(a(n)) = A001222(a(n)).

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

A257992(a(n)) = A162641(a(n)) = 0.