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%I #82 Mar 18 2024 11:49:55
%S 42,1316,131080256
%N Integers k such that k is equal to the sum of the nonprime proper divisors of k.
%C The number 37778715690312487141376 is also in the sequence. - _Daniel Suteu_, Jan 27 2020
%C The first 3 terms have the form (2^p-1)*(2^(p-1))*((2^p-1)^2-2), i.e., a Perfect number times a Carol prime. - _G. L. Honaker, Jr._, Jan 27 2020
%C In other words, the values of p are given by the intersection of A091515 and A000043. Currently, only four such values of p are known: {2, 3, 7, 19}. - _Daniel Suteu_, Jan 27 2020
%C From _Bernard Schott_, Jan 28 2020: (Start)
%C Proposition: If a number N_p is of the form Q_p * C_p where Q_p = (2^(p-1)) * (2^p - 1) is a perfect number and C_p = (2^p-1)^2-2 is a Carol prime then, the sum of the nonprime proper divisors of N_p called S_p(N_p) is equal to N_p.
%C Proof:
%C The sum of the nonprime proper divisors of N_p is:
%C S_p(N_p) = (2* Q_p - 2 - (2^p-1)) + ((Q_p - 1) * C_p).
%C In the first parenthesis, there is the sum of the nonprime proper divisors of N_p coming only from the perfect number Q_p, then in the second parenthesis, there is the sum of the nonprime proper divisors of N_p coming from C_p.
%C Then, this sum of the nonprime proper divisors of N_p, S_p(N_p) is indeed equal to N_p = (2^(p-1)) * (2^p-1) * ((2^p-1)^2-2).
%C Hence, (2^19-1)*(2^(19-1))*((2^19-1)^2-2) = 37778715690312487141376 is a term. (End)
%C 10^13 < a(4) <= 72872313094554244192 = 2^5 * 109 * 151 * 65837 * 2101546957. - _Giovanni Resta_, Jan 28 2020
%H Chris K. Caldwell and G. L. Honaker, Jr., <a href="https://primes.utm.edu/curios/page.php?curio_id=37395">Prime Curio for 42</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/42_(number)">42 (number)</a>
%e 42 is a term because 42 = 1 + 6 + 14 + 21.
%e 1316 is a term because 1316 = 1 + 4 + 14 + 28 + 94 + 188 + 329 + 658.
%t fun[p_, e_] := (p^(e+1) - 1)/(p - 1); npsigma[n_] := Times @@ fun @@@ (f = FactorInteger[n]) - Plus @@ First /@ f;; Select[Range[2, 1500], npsigma[#] == 2# &] (* _Amiram Eldar_, Jan 26 2020 *)
%o (PARI) isok(n) = sigma(n) - n - vecsum(factor(n)[,1]) == n; \\ _Daniel Suteu_, Jan 27 2020
%Y Cf. A001065, A018252, A023890, A331858.
%Y Cf. A000043, A091515, A091516 (Carol primes).
%K nonn,bref,more
%O 1,1
%A _G. L. Honaker, Jr._, Jan 26 2020
%E a(2) from Chuck Gaydos
%E a(3) from _Amiram Eldar_, Jan 26 2020
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