cp's OEIS Frontend

A025474 Exponent of the n-th prime power A000961(n).

%I A025474 #29 Aug 16 2024 08:37:15
%S A025474 0,1,1,2,1,1,3,2,1,1,4,1,1,1,2,3,1,1,5,1,1,1,1,2,1,1,1,6,1,1,1,1,4,1,
%T A025474 1,1,1,1,1,1,1,2,3,1,7,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A025474 5,1,8,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,3,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A025474 Exponent of the n-th prime power A000961(n).
%C A025474 a(n) is the number of automorphisms on the field with order A000961(n).  This group of automorphisms is cyclic of order a(n). - _Geoffrey Critzer_, Feb 23 2018
%H A025474 Reinhard Zumkeller, <a href="/A025474/b025474.txt">Table of n, a(n) for n = 1..10000</a>
%F A025474 a(n) = A100995(A000961(n)).
%F A025474 A000961(n) = A025473(n)^a(n); A056798(n) = A025473(n)^(2*a(n));
%F A025474 A192015(n) = a(n)*A025473(n)^(a(n)-1). - _Reinhard Zumkeller_, Jun 24 2011
%F A025474 a(n) = A001222(A000961(n)). - _David Wasserman_, Feb 16 2006
%t A025474 Prepend[Table[ FactorInteger[q][[1, 2]], {q,
%t A025474 Select[Range[1, 1000], PrimeNu[#] == 1 &]}], 0] (* _Geoffrey Critzer_, Feb 23 2018 *)
%o A025474 (Haskell)
%o A025474 a025474 = a001222 . a000961 -- _Reinhard Zumkeller_, Aug 13 2013
%o A025474 (PARI) A025474_upto(N)=apply(bigomega, A000961_list(N)) \\ _M. F. Hasler_, Jun 16 2022
%o A025474 (Python) A025474_upto = lambda N: [A001222(n) for n in A000961_list(N)] # _M. F. Hasler_, Jun 16 2022
%o A025474 (Python)
%o A025474 from sympy import prime, integer_nthroot, factorint
%o A025474 def A025474(n):
%o A025474     if n == 1: return 0
%o A025474     def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
%o A025474     m, k = n, f(n)
%o A025474     while m != k:
%o A025474         m, k = k, f(k)
%o A025474     return list(factorint(m).values())[0] # _Chai Wah Wu_, Aug 15 2024
%Y A025474 Cf. A000961 (the prime powers), A025473 (prime root of these), A100995 (exponent of prime powers or 0 otherwise), A001222 (bigomega), A056798 (prime powers with even exponents).
%Y A025474 Cf. A117331.
%K A025474 easy,nonn
%O A025474 1,4
%A A025474 _David W. Wilson_
%E A025474 Edited by _M. F. Hasler_, Jun 16 2022