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%I #22 Sep 29 2019 10:19:03
%S 28,145,901,1081,1749715,153533665,159114735,182475267415
%N In base 6, the reversal of n equals the sum of the reversals of the proper divisors of n.
%C 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.
%C a(6) > 10^8. - _Amiram Eldar_, Sep 28 2019
%C a(9) > 2*10^11. - _Giovanni Resta_, Sep 29 2019
%H J. Pe, <a href="http://www.numeratus.net/enlightened/pictureperfect.html">The Picture-Perfect Numbers</a>
%e 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.
%t 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}]
%K base,nonn,more
%O 1,1
%A _Joseph L. Pe_, Apr 24 2002
%E Corrected a link. - _Alan T. Koski_, Nov 25 2012
%E a(5) from _Amiram Eldar_, Sep 28 2019
%E a(6)-a(8) from _Giovanni Resta_, Sep 29 2019
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