cp's OEIS Frontend

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

%I A100995 #50 Jan 10 2025 04:30:12
%S A100995 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,
%T A100995 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,
%U A100995 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
%N A100995 If n is a prime power p^m, m >= 1, then m, otherwise 0.
%C A100995 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
%C A100995 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
%C A100995 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
%H A100995 Daniel Forgues, <a href="/A100995/b100995.txt">Table of n, a(n) for n = 1..100000</a>
%F A100995 A100994(n) = A014963(n)^a(n);
%F A100995 a(A000961(n)) = A025474(n).
%F A100995 a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * bigomega(d). - _Ilya Gutkovskiy_, Apr 15 2021
%p A100995 f:= proc(n) local F;
%p A100995     F:= ifactors(n)[2];
%p A100995     if nops(F) = 1 then F[1][2]
%p A100995     else 0
%p A100995     fi
%p A100995 end proc:
%p A100995 map(f, [$1..100]); # _Robert Israel_, Jun 09 2015
%t A100995 ppm[n_]:=If[PrimePowerQ[n],FactorInteger[n][[1,2]],0]; Array[ppm,110] (* _Harvey P. Dale_, Mar 03 2014 *)
%t A100995 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}];
%t A100995 Numerator[a]*Denominator[a] (* _Mats Granvik_, Jun 09 2015 *)
%t A100995 a = FullSimplify[Table[MangoldtLambda[n]/Log[n], {n, 1, 105}]]
%t A100995 Numerator[a]*Denominator[a] (* _Mats Granvik_, Jun 09 2015 *)
%o A100995 (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 */
%o A100995 (PARI) {a(n) = my(t); if( n<1, 0, if( t = isprimepower(n), t))} /* _Michael Somos_, Aug 15 2012 */
%o A100995 (Haskell)
%o A100995 a100995 n = f 0 n where
%o A100995    f e 1 = e
%o A100995    f e x = if r > 0 then 0 else f (e + 1) x'
%o A100995            where (x', r) = divMod x p
%o A100995    p = a020639 n
%o A100995 -- _Reinhard Zumkeller_, Mar 19 2013
%Y A100995 Cf. A028233, A069513, A010055.
%K A100995 nonn
%O A100995 1,4
%A A100995 _Reinhard Zumkeller_, Nov 26 2004
%E A100995 Edited by _Daniel Forgues_ and _N. J. A. Sloane_, Aug 18 2009