A247213 - OEIS

%I #20 Oct 13 2017 21:38:03

%S 1,2,4,8,16,32,64,105,128,210,256,315,420,512,630,840,1024,1260,1575,

%T 1680,2048,2520,3150,3360,4096,5040,6300,6720,8192,10080,11025,12600,

%U 13440,16384,20160,22050,25200,26880,32768,33075,40320,44100,50400,53760,65536

%N Numbers n = Product_(p_i^e_i) such that nn = Product_((p_i + 2)^e_i) is divisible by n.

%C That is, numbers n, such that A166590(n) is divisible by n.

%C A000079, powers of 2, is a subsequence.

%C _Thomas Ordowski_ remarks that the only squarefrees of this sequence are: 1, 2, 105, and 210.

%H Chai Wah Wu, <a href="/A247213/b247213.txt">Table of n, a(n) for n = 1..164</a>

%e A166590(2)=4 is divisible by 2, so 2 is in the sequence.

%e A166590(105) = A166590(3*5*7) = 5*7*9 = 3*(3*5*7), so 105 is in the sequence.

%t a247213[n_] := Select[Range@n, Mod[Times @@ Power @@@ Transpose[{Plus[First /@ FactorInteger@#, 2], Last /@ FactorInteger@#}], #] == 0 &]; a247213[2^16] (* _Michael De Vlieger_, Jan 07 2015 *)

%o (PARI) isok(n) = { f = factor(n); for (i=1, #f~, f[i,1] += 2); newn = factorback(f);  newn % n == 0;}

%o (Python)

%o from operator import mul

%o from functools import reduce

%o from sympy import factorint

%o A247213_list = [n for n in range(1,10**4) if n <= 1 or not reduce(mul,[(p+2)**e for p,e in factorint(n).items()]) % n]

%o # _Chai Wah Wu_, Jan 05 2015

%Y Cf. A000079, A166590.

%K nonn

%O 1,2

%A _Michel Marcus_, after a suggestion from _Thomas Ordowski_, Nov 26 2014