cp's OEIS Frontend

A069942 Reversal of n equals the sum of the reversals of the proper divisors of n.

%I A069942 #55 Sep 03 2025 09:16:34
%S A069942 6,10311,21661371,1460501511,7980062073,79862699373,798006269373
%N A069942 Reversal of n equals the sum of the reversals of the proper divisors of n.
%C A069942 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 A069942 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 A069942 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 A069942 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 A069942 Joseph L. Pe, <a href="https://vixra.org/abs/2503.0165">On a Generalization of Perfect Numbers</a>, J. Rec. Math., 31(3) (2002-2003), 168-172.
%H A069942 Joseph L. Pe, <a href="http://jlpe.tripod.com/ppn/ppn.htm">The Picture-Perfect Numbers</a>, Mathematical Spectrum, 40(1) (2007/2008).
%H A069942 Joseph L. Pe, <a href="http://www.numeratus.net/enlightened/pictureperfect.html">The Picture-Perfect Numbers</a>
%H A069942 Joseph L. Pe, <a href="http://jlpe.tripod.com/ppn/ppnpaper.htm">Picture-Perfect Numbers and Other Digit-Reversal Diversions</a>
%H A069942 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 A069942 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 A069942 f = IntegerReverse; Do[If[f[n] == Apply[Plus, Map[f, Drop[Divisors[n], -1]]], Print[n]], {n, 2, 10^8}]
%o A069942 (Python)
%o A069942 from sympy import divisors
%o A069942 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 A069942 Cf. A003601, A005117.
%K A069942 base,nice,nonn,more,changed
%O A069942 1,1
%A A069942 _Joseph L. Pe_, Apr 26 2002
%E A069942 a(5)-a(7) found by _Jens Kruse Andersen_, May 01 2002; Jul 04 2002
%E A069942 Corrected links. - _Alan T. Koski_, Nov 25 2012