Problems & Puzzles: Puzzles

 Puzzle 1108 Integers equal to its sum of nonprime proper divisors. G. L. Honaker is the author of the following Curio: 42 is the smallest number k that is equal to the sum of the nonprime proper divisors of k, i.e., 42 = 1 + 6 + 14 + 21. Q. Send your largest integer of this type?

During the week 23-29, October 2022, contributions came from Emmanuel Vantieghem, Gennady Gusev, Michael Branicky, Giorgos Kalogeropoulos, J-M Rebert, Oscar Volpatti

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Emmanuel wrote:

This is my biggest : 37778715690312487141376...

I also found the smaller solutions  1316 and 131080256.
All the solutions were of the form
(2^u)*p*q
with  p, q  primes, p = 2^(u+1) - 1 (a Mersenne prime) and  q = p^2 - 2.
It is easy to show that such numbers always will give solutions.
However, I examined the 51 known Mersenne primes.  Only four exponents lead to a solution.
For the exponents  20996011  and ,30402457  I could not find out whether or not they lead to a solution.

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In the interval up to 5*10^11 two more numbers were found:
1316=1+2+4+7+14+28+47+94+188+329+658-(2+7+47) and
131080256=1+2+4+8+16+32+64+127+2048129+8192516+16127+254+508+1016+2032+4064+8128
+32254+64508+129016+258032+516064+1032128+4096258+16385032+32770064+65540128-(127+16127+2)

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Michael wrote:

Solutions include 42, 1316, 131080256, 72872313094554244192, and 37778715690312487141376.

These are from OEIS A331805.  There, it is proved that numbers of the following form are solutions:
((2^p-1)*(2^(p-1))) * ((2^p-1)^2-2), i.e., a perfect number times a Carol prime.  The only known such

p = 2, 3, 7, 19, corresponding to four of the solutions above.  I searched p through 37699 and found
no other terms of that form.

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Giorgos wrote:

The largest integer that I could find is 37778715690312487141376
I discovered a formula that produces these numbers:
2^(s-1)*(2^s-1)*(2^(2*s)-2^(s+1)-1) where s is Mersenne prime exponent.

So, for s={2,3,7,19} we get the numbers 42, 1316, 131080256 and 37778715690312487141376

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Jean-Marc wrote:

Q. Send your largest integer of this type?

k d-digit number of this type.

d k
2 42
4 1316
9 131080256
42 = 1 + 6 + 14 + 21,
sum of the 4 nonprime proper divisors of the 2-digit number 42.

1316 = 1 + 4 + 14 + 28 + 94 + 188 + 329 + 658,
sum of the 8 nonprime proper divisors of the 4-digit number 1316.

131080256 = 1 + 4 + 8 + 16 + 32 + 64 + 254 + 508 + 1016 + 2032 + 4064 + 8128 + 32254 + 64508 + 129016 + 258032 + 516064 + 1032128 + 2048129 + 4096258 + 8192516 + 16385032 + 32770064 + 65540128,
sum of the 24 nonprime proper divisors of the 9-digit number 131080256.

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Oscar wrote:

I checked all integers below 5*10^9, finding three solutions (including the given example):
42, 1316, 131080256.
All numbers found are of the form (2^x)*y*z:
42 = (2^1)*3*7
1316 = (2^2)*7*47
131080256 = (2^6)*127*16127

So I searched for more numbers of this form, finding two more solutions with 23 and 191 digits respectively
23 digits:
37778715690312487141376
x = 19,
y = 524287,
z = 274876858367,

191 digits:
32416462582039000178559231887370949653639433739473369033778464091825269956550612948759686504
38042231430857265198986011250222927161470967422661211479522036932544288078682137674058638568
0891904

x = 168,

y = 748288838313422294120286634350736906068673408108079
z = 1157862750038882396036988194127206933757705342171831224337063368312737658092975750870
51471

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