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

6, 10311, 21661371, 1460501511, 7980062073, 79862699373, 798006269373

Offset

1,1

Comments

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.)

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.

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.

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

Links

Table of n, a(n) for n=1..7.

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

Joseph L. Pe, The Picture-Perfect Numbers, Mathematical Spectrum, 40(1) (2007/2008).

Joseph L. Pe, The Picture-Perfect Numbers

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

Rulthan P. Sumicad, On the Picture-Perfect Number, J. Math. Stat. Studies (2023).

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

Cf. A003601, A005117.

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