|
%I #42 Dec 08 2023 12:05:52
%S 6,10311,21661371,1460501511,7980062073,79862699373,798006269373
%N Reversal of n equals the sum of the reversals of the proper divisors of n.
%C These numbers are called picture-perfect numbers (ppn's). If a ppn is placed on one side of an equal sign and its proper divisors on the other side, then the resulting equation read backwards is valid. The first three ppn's were found by _Joseph L. Pe_. The fourth ppn was discovered by Daniel Dockery. Mark Ganson conjectures that every ppn is divisible by 3. (Compare this with the still unresolved conjecture that every perfect number is divisible by 2.)
%C Jens Kruse Andersen discovered the remarkable result that if the decimal number p = 140z10n89 is prime, then the product 57p is picture-perfect and conversely, where z is any number (possibly none) of 0's and n is any number (possibly none) of 9's.
%C Andersen has recently found the following extension of his result: If p=140{(0)_z10(9)_n89}_k is prime, then 3*19*p is a ppn and conversely. Here (0)_z is a string of z=>0 "zeros", (9)_n is a string of n=>0 "nines", k is the number of repetitions of the part {(0)_z10(9)_n89} with varying numbers of zeros and nines in each repetition.
%C It seems this sequence is a subsequence of A003601 (arithmetic numbers) and A005117 (squarefree numbers). This is trivially true for the sequence of Andersen's numbers (of the form 3*19*p). - _Ivan N. Ianakiev_, Feb 12 2023
%H Joseph L. Pe, <a href="http://www.numeratus.net/fperfect/fperfect.html">On a Generalization of Perfect Numbers</a>, J. Rec. Math., 31(3) (2002-2003), 168-172.
%H Joseph L. Pe, <a href="http://jlpe.tripod.com/ppn/ppn.htm">The Picture-Perfect Numbers</a>, Mathematical Spectrum, 40(1) (2007/2008).
%H Joseph L. Pe, <a href="http://www.numeratus.net/enlightened/pictureperfect.html">The Picture-Perfect Numbers</a>
%H Joseph L. Pe, <a href="http://jlpe.tripod.com/ppn/ppnpaper.htm">Picture-Perfect Numbers and Other Digit-Reversal Diversions</a>
%H Rulthan P. Sumicad, <a href="https://doi.org/10.32996/jmss.2023.4.4.11">On the Picture-Perfect Number</a>, J. Math. Stat. Studies (2023).
%e The reversal of 10311 is 11301 and the reversals of its proper divisors are: 1, 3, 7, 12, 194, 3741, 7343. Adding the proper divisor reversals 1 + 3 + 7 + 12 + 194 + 3741 + 7343 = 11301, so 10311 belongs to the sequence.
%t f = IntegerReverse; Do[If[f[n] == Apply[Plus, Map[f, Drop[Divisors[n], -1]]], Print[n]], {n, 2, 10^8}]
%o (Python)
%o from sympy import divisors
%o A069942 = [n for n in range(1,10**5) if sum(list(map(lambda x: int(str(x)[::-1]) if x < n else 0, divisors(n)))) == int(str(n)[::-1])] # _Chai Wah Wu_, Aug 13 2014
%Y Cf. A003601, A005117.
%K base,nice,nonn
%O 1,1
%A _Joseph L. Pe_, Apr 26 2002
%E a(5)-a(7) found by _Jens Kruse Andersen_, May 01, 2002; Jul 04 2002
%E Corrected links. - _Alan T. Koski_, Nov 25 2012
| 