A010055 1 if n is a prime power p^k (k >= 0), otherwise 0.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Characteristic function of unit or prime powers p^k (k >= 1). Characteristic function of prime powers p^k (k >= 0). - Daniel Forgues, Mar 03 2009

See A065515 for partial sums. - Reinhard Zumkeller, Nov 22 2009

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Index entries for characteristic functions

Crossrefs

Cf. A069513 (1 if n is a prime power p^k (k >= 1), else 0.)

Cf. A268340.

Cf. A100995.

Cf. A001221, A000961, A024619, A065515,

Programs

Haskell

a010055 n = if a001221 n <= 1 then 1 else 0
-- Reinhard Zumkeller, Nov 28 2015, Mar 19 2013, Nov 17 2011

Maple

A010055 := proc(n)
    if n =1 then
        1;
    else
        if nops(ifactors(n)[2]) = 1 then
            1;
        else
            0 ;
        end if;
    end if;
end proc: # R. J. Mathar, May 25 2017

Mathematica

A010055[n_]:=Boole[PrimeNu[n]<=1]; A010055/@Range[20] (* Enrique Pérez Herrero, May 30 2011 *)
{1}~Join~Table[Boole@ PrimePowerQ@ n, {n, 2, 105}] (* Michael De Vlieger, Feb 02 2016 *)

PARI

for(n=1,120,print1(omega(n)<=1,","))

Python

from sympy import primefactors
def A010055(n): return int(len(primefactors(n)) <= 1) # Chai Wah Wu, Mar 31 2023

Formula

Dirichlet generating function: 1 + ppzeta(s). Here ppzeta(s) = Sum_{p prime} Sum_{k>=1} 1/(p^k)^s. Note that ppzeta(s) = Sum_{p prime} 1/(p^s-1) = Sum_{k>=1} primezeta(k*s). - Franklin T. Adams-Watters, Sep 11 2005

a(n) = 0^(A119288(n)-1). - Reinhard Zumkeller, May 13 2006

a(A000961(n)) = 1; a(A024619(n)) = 0. - Reinhard Zumkeller, Nov 17 2011

a(n) = if A001221(n) <= 1 then 1, otherwise 0. - Reinhard Zumkeller, Nov 28 2015

If n >= 2, a(n) = A069513(n). - Jeppe Stig Nielsen, Feb 02 2016

Conjecture: a(n) = (n - A048671(n))/A000010(n) for all n > 1. - Velin Yanev, Mar 10 2021 [The conjecture is true. - Andrey Zabolotskiy, Mar 11 2021]

Extensions

More terms from Charles R Greathouse IV, Mar 12 2008

Edited by Daniel Forgues, Mar 02 2009

Comment re Galois fields moved to A069513 by Franklin T. Adams-Watters, Nov 02 2009