A351979 Numbers whose prime factorization has all odd prime indices and all even prime exponents.

1, 4, 16, 25, 64, 100, 121, 256, 289, 400, 484, 529, 625, 961, 1024, 1156, 1600, 1681, 1936, 2116, 2209, 2500, 3025, 3481, 3844, 4096, 4489, 4624, 5329, 6400, 6724, 6889, 7225, 7744, 8464, 8836, 9409, 10000, 10609, 11881, 12100, 13225, 13924, 14641, 15376

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 odd parts and all even multiplicities, counted by A035457 (see Emeric Deutsch's comment there).

Examples

			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
		

Links

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

Crossrefs

The second condition alone (exponents all even) is A000290, counted by A035363.

The distinct prime factors of terms all come from A031368.

These partitions are counted by A035457 or A000009 aerated.

The first condition alone (indices all odd) is A066208, counted by A000009.

The squarefree square roots are A258116, even A258117.

A056166 = exponents all prime, counted by A055923.

A066207 = indices all even, counted by complement of A086543.

A076610 = indices all prime, counted by A000607.

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.

A268335 = exponents all odd, counted by A055922.

A325131 = disjoint indices from exponents, counted by A114639.

A346068 = indices and exponents all prime, counted by A351982.

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

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

A352142 = odd indices and odd multiplicities, counted by A117958.

Cf. A000720, A028260, A045931, A055396, A061395, A106529, A181819, A195017, A276078, A324588, A325698, A325700.

Programs

Mathematica

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

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

Formula

Squares of elements of A066208.

Intersection of A066208 and A000290.

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

A162641(a(n)) = A001221(a(n)).

A162642(a(n)) = A257992(a(n)) = 0.

Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(2*k-1)^2) = 1.4135142... . - Amiram Eldar, Sep 19 2022