A036896 Odd refactorable numbers.

1, 9, 225, 441, 625, 1089, 1521, 2025, 2601, 3249, 4761, 5625, 6561, 7569, 8649, 12321, 15129, 16641, 19881, 25281, 31329, 33489, 35721, 40401, 45369, 47961, 50625, 56169, 62001, 71289, 84681, 91809, 95481, 99225, 103041, 106929, 114921

Offset

1,2

Comments

Odd refactorable numbers are always squares.

All terms = 1 (mod 8). [Zak Seidov, May 25 2010]

Links

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1001 terms from Harvey P. Dale)

S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.

S. Colton, HR - Automatic Theory Formation in Pure Mathematics

Example

9 is refactorable because tau(9)=3 and 3 divides 9.

Mathematica

Do[If[IntegerQ[n/DivisorSigma[0, n]], Print[n]], {n, 1, 100000, 2}]
Select[Range[1, 1001, 2]^2, Divisible[#, DivisorSigma[0, #]]&] (* Harvey P. Dale, Jan 22 2012 *)

Prog

(PARI) is(n)=n%2&&issquare(n)&&n%numdiv(n)==0 \\ Charles R Greathouse IV, Apr 23 2013
(PARI) list(lim)=my(v=List(), f); forstep(n=1, sqrtint(lim\1), 2, f=factor(n)[, 2]; if(n^2%prod(i=1, #f, 2*f[i]+1)==0, listput(v, n^2))); Vec(v) \\ Charles R Greathouse IV, Apr 23 2013
(Python)
from itertools import count, islice
from math import prod
from sympy import factorint
def A036896_gen(): # generator of terms
    for n in count(1, 2):
        if not (m:=n**2)%prod(e<<1|1 for e in factorint(n).values()): yield m
A036896_list = list(islice(A036896_gen(), 40)) # Chai Wah Wu, Oct 04 2024

Crossrefs

Subsequence of A033950 and of A016754.

Sequence in context: A264848 A033632 A110260 * A120319 A057530 A014736

Adjacent sequences: A036893 A036894 A036895 * A036897 A036898 A036899

Keyword

nonn,easy

Author

Simon Colton (simonco(AT)cs.york.ac.uk)

Status

approved