

A069942


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


12




OFFSET

1,1


COMMENTS

These numbers are called pictureperfect 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.)
Jens Kruse Andersen discovered the remarkable result that if the decimal number p = 140z10n89 is prime, then the product 57p is pictureperfect and conversely, where z is any number (possibly none) of 0's and n is any number (possibly none) of 9's.
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.


LINKS

Table of n, a(n) for n=1..7.
Joseph L. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (20022003), 168172.
Joseph L. Pe, The PicturePerfect Numbers, Mathematical Spectrum, 40(1) (2007/2008).
Joseph L. Pe, The PicturePerfect Numbers
Joseph L. Pe, PicturePerfect Numbers and Other DigitReversal Diversions


EXAMPLE

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.


MATHEMATICA

f = IntegerReverse; Do[If[f[n] == Apply[Plus, Map[f, Drop[Divisors[n], 1]]], Print[n]], {n, 2, 10^8}]


PROG

(Python)
from sympy import divisors
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


CROSSREFS

Sequence in context: A329911 A088021 A102979 * A261823 A146202 A227889
Adjacent sequences: A069939 A069940 A069941 * A069943 A069944 A069945


KEYWORD

base,nice,nonn


AUTHOR

Joseph L. Pe, Apr 26 2002


EXTENSIONS

a(5)a(7) found by Jens Kruse Andersen, May 01, 2002; Jul 04 2002
Corrected links.  Alan T. Koski, Nov 25 2012


STATUS

approved



