A069917 In base 6, the reversal of n equals the sum of the reversals of the proper divisors of n.

28, 145, 901, 1081, 1749715, 153533665, 159114735, 182475267415

Offset

1,1

Comments

A number n is called "picture-perfect" if the reversal of n equals the sum of the reversals of the proper divisors of n. These base-6 picture-perfect numbers were found by Mark Ganson while searching for (base-10) picture-perfect numbers. He observes that the digital sum of their base-10 representations = 10 and conjectures that this is the case for all base-6 picture-perfect numbers. The only (base-10) picture-perfect numbers not exceeding 1.3 * 10^9 are 6, 10311 and 21661371.

a(6) > 10^8. - Amiram Eldar, Sep 28 2019

a(9) > 2*10^11. - Giovanni Resta, Sep 29 2019

Links

Table of n, a(n) for n=1..8.

J. Pe, The Picture-Perfect Numbers

Example

28 has proper divisors 1, 2, 4, 7, 14. 28 = 44_6, 1 = 1_6, 2 = 2_6, 4 = 4_6, 7 = 11_6, 14 = 22_6. Reversing these base-6 numbers, we have 44_6 = 1_6 + 2_6 + 4_6 + 11_6 + 22_6 so 28 belongs to the sequence.

Mathematica

base=6; f[n_] := FromDigits[Reverse[IntegerDigits[n, base]], base]; baseDivisors[n_, base_] := IntegerDigits[Drop[Divisors[n], -1], base]; Do[ startFrom = 2; Do[If[f[n] == Apply[Plus, Map[f, Drop[Divisors[n], -1]]], Print["base = ", base, ", n = ", n, ") ", IntegerDigits[n, base], " divisors: ", Drop[Divisors[n], -1], " base divisors: ", baseDivisors[n, base]]], {n, startFrom, 10000}], {base, 2, 10}]

Crossrefs

Sequence in context: A042530 A042532 A187608 * A028380 A219887 A271636

Adjacent sequences: A069914 A069915 A069916 * A069918 A069919 A069920

Keyword

base,nonn,more

Author

Joseph L. Pe, Apr 24 2002

Extensions

Corrected a link. - Alan T. Koski, Nov 25 2012

a(5) from Amiram Eldar, Sep 28 2019

a(6)-a(8) from Giovanni Resta, Sep 29 2019

Status

approved