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 A100995 If n is a prime power p^m, m >= 1, then m, otherwise 0. 31
 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 (list; graph; refs; listen; history; text; internal format)
 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 Daniel Forgues, Table of n, a(n) for n = 1..100000 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 Cf. A028233, A069513, A010055. Sequence in context: A171963 A292131 A255740 * A319273 A329615 A272894 Adjacent sequences:  A100992 A100993 A100994 * A100996 A100997 A100998 KEYWORD nonn AUTHOR Reinhard Zumkeller, Nov 26 2004 EXTENSIONS Edited by Daniel Forgues and N. J. A. Sloane, Aug 18 2009 STATUS approved

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Last modified July 4 11:49 EDT 2022. Contains 355075 sequences. (Running on oeis4.)