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A036896 Odd refactorable numbers.

%I A036896 #36 Oct 04 2024 20:48:46
%S A036896 1,9,225,441,625,1089,1521,2025,2601,3249,4761,5625,6561,7569,8649,
%T A036896 12321,15129,16641,19881,25281,31329,33489,35721,40401,45369,47961,
%U A036896 50625,56169,62001,71289,84681,91809,95481,99225,103041,106929,114921
%N A036896 Odd refactorable numbers.
%C A036896 Odd refactorable numbers are always squares.
%C A036896 All terms = 1 (mod 8). [_Zak Seidov_, May 25 2010]
%H A036896 Giovanni Resta, <a href="/A036896/b036896.txt">Table of n, a(n) for n = 1..10000</a> (first 1001 terms from Harvey P. Dale)
%H A036896 S. Colton, <a href="http://www.cs.uwaterloo.ca/journals/JIS/colton/joisol.html">Refactorable Numbers - A Machine Invention</a>, J. Integer Sequences, Vol. 2, 1999, #2.
%H A036896 S. Colton, <a href="http://web.archive.org/web/20070831060523/http://www.dai.ed.ac.uk/homes/simonco/research/hr/">HR - Automatic Theory Formation in Pure Mathematics</a>
%e A036896 9 is refactorable because tau(9)=3 and 3 divides 9.
%t A036896 Do[If[IntegerQ[n/DivisorSigma[0, n]], Print[n]], {n, 1, 100000, 2}]
%t A036896 Select[Range[1,1001,2]^2,Divisible[#,DivisorSigma[0,#]]&] (* _Harvey P. Dale_, Jan 22 2012 *)
%o A036896 (PARI) is(n)=n%2&&issquare(n)&&n%numdiv(n)==0 \\ _Charles R Greathouse IV_, Apr 23 2013
%o A036896 (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
%o A036896 (Python)
%o A036896 from itertools import count, islice
%o A036896 from math import prod
%o A036896 from sympy import factorint
%o A036896 def A036896_gen(): # generator of terms
%o A036896     for n in count(1,2):
%o A036896         if not (m:=n**2)%prod(e<<1|1 for e in factorint(n).values()): yield m
%o A036896 A036896_list = list(islice(A036896_gen(),40)) # _Chai Wah Wu_, Oct 04 2024
%Y A036896 Subsequence of A033950 and of A016754.
%K A036896 nonn,easy
%O A036896 1,2
%A A036896 Simon Colton (simonco(AT)cs.york.ac.uk)