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

1,4

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

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

Mathematica

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

Prog

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

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.

Sequence in context: A348285 A164953 A136622 * A136575 A309898 A193592

Adjacent sequences: A025471 A025472 A025473 * A025475 A025476 A025477

Keyword

easy,nonn

Author

David W. Wilson

Extensions

Edited by M. F. Hasler, Jun 16 2022

Status

approved