Voozh

A247213

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

1, 2, 4, 8, 16, 32, 64, 105, 128, 210, 256, 315, 420, 512, 630, 840, 1024, 1260, 1575, 1680, 2048, 2520, 3150, 3360, 4096, 5040, 6300, 6720, 8192, 10080, 11025, 12600, 13440, 16384, 20160, 22050, 25200, 26880, 32768, 33075, 40320, 44100, 50400, 53760, 65536

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

A000079, powers of 2, is a subsequence.

Thomas Ordowski remarks that the only squarefrees of this sequence are: 1, 2, 105, and 210.

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..164

EXAMPLE

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

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

MATHEMATICA

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 *)

PROG

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

(Python)

from operator import mul

from functools import reduce

from sympy import factorint

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]

# Chai Wah Wu, Jan 05 2015

CROSSREFS

Cf. A000079, A166590.

Sequence in context: A008881 A208743 A335853 * A302934 A069050 A343844

Adjacent sequences: A247210 A247211 A247212 * A247214 A247215 A247216