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

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, 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, 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

Offset: 1

Author

David W. Wilson

Keywords

• easy
• nonn

Comments

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

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).

Cf. A117331.

Programs

Haskell

a025474 = a001222 . a000961 -- Reinhard Zumkeller, Aug 13 2013

Mathematica

Prepend[Table[ FactorInteger[q][[1, 2]], {q,
Select[Range[1, 1000], PrimeNu[#] == 1 &]}], 0] (* Geoffrey Critzer, Feb 23 2018 *)

PARI

A025474_upto(N)=apply(bigomega, A000961_list(N)) \\ M. F. Hasler, Jun 16 2022

Python

A025474_upto = lambda N: [A001222(n) for n in A000961_list(N)] # M. F. Hasler, Jun 16 2022

Python

from sympy import prime, integer_nthroot, factorint
def A025474(n):
    if n == 1: return 0
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return list(factorint(m).values())[0] # Chai Wah Wu, Aug 15 2024

Formula

a(n) = A100995(A000961(n)).

A000961(n) = A025473(n)^a(n); A056798(n) = A025473(n)^(2*a(n));

A192015(n) = a(n)*A025473(n)^(a(n)-1). - Reinhard Zumkeller, Jun 24 2011

a(n) = A001222(A000961(n)). - David Wasserman, Feb 16 2006

Extensions

Edited by M. F. Hasler, Jun 16 2022