A091862 a(n) = 1 if the sum of all exponents of the prime-factorization of n has no carries when summed in base 2, or a(n) = 0 if there are any carries.

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

Offset: 1

Author

Leroy Quet, Mar 13 2004

Keywords

• nonn
• base

Comments

Characteristic function for A268375. - Antti Karttunen, Nov 23 2017

Examples

			a(12) = 1 because 12 = 2^2 *3^1 and, in base 2, 2 = '10', 1 = '1' and '10' and '1' have their ones in different positions. But a(24) = 0 because 24 = 2^3 *3^1 and in base 2 3 = '11', 1 = '1', which both share a rightmost one.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for characteristic functions.
• Index entries for sequences computed from exponents in factorization of n.
• Index entries for sequences related to binary expansion of n.

Crossrefs

Cf. A091863, A267116, A268374, A289618.

Cf. A268375 (positions of ones), A268376 (of zeros).

Programs

Mathematica

f[e_] := Position[Reverse[IntegerDigits[e, 2]], 1] // Flatten; a[n_] := Boole[UnsameQ @@ Flatten[f /@ FactorInteger[n][[;; , 2]]]]; Array[a, 100] (* Amiram Eldar, Dec 23 2023 *)

PARI

a(n) = {my(e = factor(n)[,2], b = 0); for(i=1, #e, b = bitor(b, e[i])); n == 1 || b == vecsum(e);} \\ Amiram Eldar, Dec 23 2023

Formula

If A268374(n) = 0, then a(n) = 1, 0 otherwise. - Antti Karttunen, Nov 23 2017

Extensions

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004