A331805 Integers k such that k is equal to the sum of the nonprime proper divisors of k.

42, 1316, 131080256

Offset

1,1

Comments

The number 37778715690312487141376 is also in the sequence. - Daniel Suteu, Jan 27 2020

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

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

From Bernard Schott, Jan 28 2020: (Start)

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.

Proof:

The sum of the nonprime proper divisors of N_p is:

S_p(N_p) = (2* Q_p - 2 - (2^p-1)) + ((Q_p - 1) * C_p).

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.

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).

Hence, (2^19-1)*(2^(19-1))*((2^19-1)^2-2) = 37778715690312487141376 is a term. (End)

10^13 < a(4) <= 72872313094554244192 = 2^5 * 109 * 151 * 65837 * 2101546957. - Giovanni Resta, Jan 28 2020

Links

Table of n, a(n) for n=1..3.

Chris K. Caldwell and G. L. Honaker, Jr., Prime Curio for 42

Wikipedia, 42 (number)

Example

42 is a term because 42 = 1 + 6 + 14 + 21.
1316 is a term because 1316 = 1 + 4 + 14 + 28 + 94 + 188 + 329 + 658.

Mathematica

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 *)

Prog

(PARI) isok(n) = sigma(n) - n - vecsum(factor(n)[, 1]) == n; \\ Daniel Suteu, Jan 27 2020

Crossrefs

Cf. A001065, A018252, A023890, A331858.

Cf. A000043, A091515, A091516 (Carol primes).

Sequence in context: A075922 A230939 A331858 * A348775 A238537 A077123

Adjacent sequences: A331802 A331803 A331804 * A331806 A331807 A331808

Keyword

nonn,bref,more

Author

G. L. Honaker, Jr., Jan 26 2020

Extensions

a(2) from Chuck Gaydos

a(3) from Amiram Eldar, Jan 26 2020

Status

approved