Problems & Puzzles:
Puzzles
Puzzle 26. Honaker & Jud. McCranie puzzle
"30103 is the only known multidigit palindromic prime found by averaging the
divisors of a composite number. [McCranie and Honaker, July 1998]
By the way they have found two solutions:
30103 = average divisors of (149645) and
30103 = average divisors of (179574)
The divisors of 149645 are 1, 5, 173, 865, 29929, and 149645.
30103 = (1+ 5+173+ 865+ 29929+149645)/6
The divisors of 179574 are 1, 2, 3, 6, 173, 346, 519, 1038, 29929,
59858, 89787, and 179574.
30103 = (1+ 2+ 3+ 6+ 173+ 346+ 519+ 1038+ 29929+59858+ 89787+179574)/12
Question : Can the average of the divisors of other composite number to produce
another palprime ?
Almost eleven years later Anton Vrba sent the first bite
to this puzzle. On May 09 he wrote:
Let AD[n] be the average of the divisors of a
composite n, and if Q is a prime then:
AD[Q^{2}]=(Q^{2}+Q+1)/3
as the divisors of Q^{2} are 1, Q and Q^{2},
and
AD[5xQ^{2}]=(Q^{2}+Q+1)
as there are 6 divisors and their sum being (1+5)+(1+5)Q+(1+5)Q^{2}
And similarly AD[6xQ^{2}]
=(Q^{2}+Q+1), 12 divisors
1,2,3,6,Q….6Q^{2}
Now we can search for palindromes that have the form
(Q^{2}+Q+1)/3 or (Q^{2}+Q+1):
30103 = AD[5x 173^{2}]
= AD[6x 173^{2}] , prime
1081801 =AD[1801^{2}],
composite
37011411073 =AD[5x 192383^{2}],
composite
159859343958951 = AD[5x
12643549^{2}], composite
131374494473131 = AD[19852543^{2}],
composite
Searched up to Q=
21780960161 so we can say:
for all palindromes less
than 1.58 x 10^{20 } , there are only five
palindromes that equal the average of the divisors of a composite
number and only one of these is prime.
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