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A100995
If n is a prime power p^m, m >= 1, then m, otherwise 0.
32
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
1,4
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 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
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
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 *)
PROG
(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 */
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Nov 26 2004
EXTENSIONS
Edited by Daniel Forgues and N. J. A. Sloane, Aug 18 2009
STATUS
approved