A005731 Related to n-th powers of polynomials: factors complementary to A005730. (Formerly M0427)

1, 1, 2, 3, 2, 5, 2, 21, 2, 3, 1, 55, 3, 13, 2, 21, 2, 85, 1, 57, 2, 1, 1, 8855, 2, 15, 2, 39, 1, 29, 10, 651, 2, 1, 2, 935, 1, 37, 2, 399, 1, 2665, 1, 129, 2, 1, 1, 416185, 2, 21, 2, 15, 1, 4505, 2, 39, 14, 1, 1, 357599, 1, 305, 2, 651, 6, 5, 1, 201, 2, 39, 1, 10687985, 2, 73, 2

Offset

1,3

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

T. Chinburg and M. Henriksen, Sums of k-th powers in the ring of polynomials with integer coefficients, Acta Arithmetica, 29 (1976), 227-250.

Formula

a(n) = A005729(n)/A005730(n).

Prog

(Python)
from itertools import count
from sympy import nextprime
def A005731(n):
    c, p = 1, 2
    while p < n:
        if n%p:
            for m in count(2):
                if (p**m-1)//(p-1) > n:
                    break
                for r in count(1):
                    q = (p**(m*r)-1)//(p**r-1)
                    if q > n:
                        break
                    if not n % q:
                        c *= p
                        break
                else:
                    continue
                if q <= n:
                    break
        p = nextprime(p)
    return c # Chai Wah Wu, Mar 10 2024

Crossrefs

Cf. A005729, A005730.

Sequence in context: A271774 A331602 A118176 * A132962 A046145 A103309

Adjacent sequences: A005728 A005729 A005730 * A005732 A005733 A005734

Keyword

nonn,changed

Author

N. J. A. Sloane

Extensions

More terms from Emeric Deutsch, Jan 24 2005

Status

approved