A353370 a(n) = 1 if the prime factorization of n contains as many even as odd prime indices, when counted with multiplicity, otherwise 0.

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1

Offset

1

Comments

This is the characteristic function of A325698, see comments there.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Index entries for characteristic functions

Index entries for sequences computed from indices in prime factorization

Formula

a(p) = 0 for all primes p.

a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.

If a(x)+a(y) > 0, then a(x*y) = a(x)*a(y).

Example

For n = 84 = 2 * 2 * 3 * 7 the condition is satisfied, because it has two instances of odd-indexed primes (A031368), namely (2*2), and also two instances (3 and 7) of the even-indexed primes (A031215), therefore a(84) = 1.

Mathematica

{1}~Join~Array[Boole[#2 == #1 - #2] & @@ {Length[#], Count[#, _?EvenQ]} &@ Flatten@ Apply[ConstantArray[PrimePi[#1], #2] &, FactorInteger[#], {1}] &, 121, 2] (* Michael De Vlieger, Apr 16 2022 *)

Prog

(PARI) A353370(n) = { my(f = factor(n)); (0==sum(i=1, #f~, f[i, 2]*((-1)^(primepi(f[i, 1])%2)))); }; \\ Antti Karttunen, Apr 16 2022
(Python)
from sympy import factorint, primepi
def a(n):
    v = [0, 0]
    for p, e in factorint(n).items(): v[primepi(p)%2] += e
    return int(v[0] == v[1])
print([a(n) for n in range(1, 123)]) # Michael S. Branicky, Apr 16 2022

Crossrefs

Characteristic function of A325698.

Cf. A003961, A031215, A031368, A348717, A353371, A353372 (inverse Möbius transform), A353373.

Cf. also A353331.

Sequence in context: A132918 A358750 A205809 * A355940 A348033 A327153

Adjacent sequences: A353367 A353368 A353369 * A353371 A353372 A353373

Keyword

nonn

Author

Antti Karttunen, Apr 16 2022

Status

approved