1,1

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 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 picture-perfect, and conversely, where z is any number (possibly none) of 0's and n is any number (possibly none) of 9s.

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.

J. Pe, On a Generalization of Perfect Numbers, J. Rec. Math., 31(3) (2002-2003), 168-172.

Pe, J., Systematic Undertaking to Find Picture-Perfect Numbers Discussion Forum

Pe, J., The Picture-Perfect Numbers

Pe, J. Picture-Perfect Numbers and Other Digit-Reversal Diversions

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.

f[n_] := FromDigits[Reverse[IntegerDigits[n]]]; Do[If[f[n] == Apply[Plus, Map[f, Drop[Divisors[n], -1]]], Print[n]], {n, 2, 10^8}]

Sequence in context: A013784 A137040 A088021 this_sequence A093897 A062782 A036773

Adjacent sequences: A069939 A069940 A069941 this_sequence A069943 A069944 A069945

base,nice,nonn

Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Apr 26 2002

a(5)-a(7) found by Jens Kruse Andersen, May 01, 2002; Jul 04 2002.