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A100995
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If n is a prime power p^m, m >= 1, then m, otherwise 0.
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31
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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
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OFFSET
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1,4
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COMMENTS
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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
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LINKS
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Daniel Forgues, Table of n, a(n) for n = 1..100000
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A028233, A069513, A010055.
Sequence in context: A171963 A292131 A255740 * A319273 A329615 A272894
Adjacent sequences: A100992 A100993 A100994 * A100996 A100997 A100998
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KEYWORD
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nonn
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AUTHOR
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Reinhard Zumkeller, Nov 26 2004
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EXTENSIONS
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Edited by Daniel Forgues and N. J. A. Sloane, Aug 18 2009
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STATUS
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approved
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