A100995 If n is a prime power p^m, m >= 1, then m, otherwise 0.

0, 1, 1, 2, 1, 0, 1, 3, 2, 0, 1, 0, 1, 0, 0, 4, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 5, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 6, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Author

Reinhard Zumkeller, Nov 26 2004

Keywords

• nonn

Comments

Calculate matrix powers: (A175992^1)/1 - (A175992^2)/2 + (A175992^3)/3 - (A175992^4)/4 + ... Then the nonzero values of a(n) are found as reciprocals in the first column. Compare this to the Taylor series for log(1+x) = (x)/1 - (x^2)/2 + (x^3)/3 - (x^4)/4 + ... Therefore it is natural to write 0, 1/1, 1/1, 1/2, 1/1, 0, 1/1, 1/3, 1/2, 0, 1/1, ... Raising n to a such power gives A014963. - Mats Granvik, Gary W. Adamson, Apr 04 2011

The Dirichlet series that generates the reciprocals of this sequence is the logarithm of the Riemann zeta function. - Mats Granvik, Gary W. Adamson, Apr 04 2011

Number of automorphisms of the finite field with n elements, or 0 if the field does not exist. For n=p^k where p is a prime and k is an integer, the automorphism group of the finite field with n elements is a cyclic group of order k generated by the Frobenius endomorphism. - Yancheng Lu, Jan 11 2021

Links

• Daniel Forgues, Table of n, a(n) for n = 1..100000

Crossrefs

Cf. A028233, A069513, A010055.

Programs

Haskell

a100995 n = f 0 n where
   f e 1 = e
   f e x = if r > 0 then 0 else f (e + 1) x'
           where (x', r) = divMod x p
   p = a020639 n
-- Reinhard Zumkeller, Mar 19 2013

Maple

f:= proc(n) local F;
    F:= ifactors(n)[2];
    if nops(F) = 1 then F[1][2]
    else 0
    fi
end proc:
map(f, [$1..100]); # Robert Israel, Jun 09 2015

Mathematica

ppm[n_]:=If[PrimePowerQ[n],FactorInteger[n][[1,2]],0]; Array[ppm,110] (* Harvey P. Dale, Mar 03 2014 *)
a=Table[Limit[Sum[If[Mod[n, k] == 0, MoebiusMu[n/k]/(n/k)^(s - 1)/(1 - 1/n^(s - 1)), 0], {k, 1, n}], s -> 1], {n, 1, 105}];
Numerator[a]*Denominator[a] (* Mats Granvik, Jun 09 2015 *)
a = FullSimplify[Table[MangoldtLambda[n]/Log[n], {n, 1, 105}]]
Numerator[a]*Denominator[a] (* Mats Granvik, Jun 09 2015 *)

PARI

{a(n) = my(t); if( n<1, 0, t = factor(n); if( [1,2] == matsize(t), t[1,2], 0))} /* Michael Somos, Aug 15 2012 */

PARI

{a(n) = my(t); if( n<1, 0, if( t = isprimepower(n), t))} /* Michael Somos, Aug 15 2012 */

Formula

A100994(n) = A014963(n)^a(n);

a(A000961(n)) = A025474(n).

a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * bigomega(d). - Ilya Gutkovskiy, Apr 15 2021

Extensions

Edited by Daniel Forgues and N. J. A. Sloane, Aug 18 2009