A353672 -id:A353672

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A353675

a(n) = 1 if n is an odd number with an even number of distinct prime factors, otherwise 0.

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for characteristic functions

FORMULA

a(n) = A000035(n) * (1-A092248(n)).

a(n) = A000035(n) - A353673(n).

a(n) >= A353676(n).

EXAMPLE

n = 45 = 3^2 * 5 is an odd number with two distinct prime factors, therefore a(45) = 1.

n = 1155 = 3*5*7*11 is an odd number with four distinct prime factors, therefore a(1155) = 1.

MATHEMATICA

Table[If[OddQ[n]&&EvenQ[PrimeNu[n]], 1, 0], {n, 130}] (* Harvey P. Dale, Feb 07 2024 *)

PROG

(PARI) A353675(n) = ((n%2) && !(omega(n)%2));

CROSSREFS

Characteristic function of {1} UNION A098905.

Cf. A000035, A001221, A092248, A353672, A353673, A353674.

After n=1 differs from A353676 for the next time at n=1155, where a(1155)=1, while A353676(1155)=0.

Cf. also A353557.

KEYWORD

nonn

AUTHOR

Antti Karttunen, May 03 2022

STATUS

approved

A353673

a(n) = 1 if n is an odd number with an odd number of distinct prime factors, otherwise 0.

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for characteristic functions

FORMULA

a(n) = A000035(n) * A092248(n).

a(n) = A000035(n) - A353675(n).

a(n) = A092248(n) - A353674(n).

EXAMPLE

n = 9 = 3^2 is an odd number with an odd number of distinct prime factors, therefore a(9) = 1.

n = 105 = 3*5*7 is an odd number with an odd number of distinct prime factors, therefore a(105) = 1.

PROG

(PARI) A353673(n) = ((n%2) && (omega(n)%2));

CROSSREFS

Characteristic function of A098903.

Cf. A000035, A001221, A092248, A353672, A353674, A353675.

Differs from A174275 for the first time at n=105, where a(105) = 1, while A174275(105) = 0.

Cf. also A353558.

KEYWORD

nonn

AUTHOR

Antti Karttunen, May 03 2022

STATUS

approved

A353674

a(n) = 1 if n is an even number with an odd number of distinct prime factors, otherwise 0.

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for characteristic functions

FORMULA

a(n) = A059841(n) * A092248(n).

a(n) = A059841(n) - A353674(n).

a(n) = A092248(n) - A353673(n).

MATHEMATICA

Table[If[EvenQ[n]&&OddQ[PrimeNu[n]], 1, 0], {n, 140}] (* Harvey P. Dale, Jul 16 2023 *)

PROG