Problems & Puzzles: Puzzles

"30103 is the only known multi-digit 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 pal-prime ?

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[Q²]=(Q²+Q+1)/3 as the divisors of Q² are 1, Q and Q², and

AD[5xQ²]=(Q²+Q+1) as there are 6  divisors and their sum being  (1+5)+(1+5)Q+(1+5)Q²

And similarly AD[6xQ²] =(Q²+Q+1), 12 divisors 1,2,3,6,Q….6Q²

Now we can search for palindromes that have the form (Q²+Q+1)/3 or (Q²+Q+1):

30103 = AD[5x 173²] =  AD[6x 173²]  , prime

1081801 =AD[1801²], composite

37011411073 =AD[5x 192383²], composite

159859343958951 = AD[5x 12643549²], composite

131374494473131 = AD[19852543²], composite

Searched up to Q= 21780960161 so we can say:

for all palindromes less than 1.58 x 10²⁰  , 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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