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Odd numbers x such that x and x + 2 are both sums of divisors, i.e., elements of A000203.
1

%I #35 Jan 02 2023 12:30:54

%S 1,13,91,241573,38152387,139415801707,55342019130181,61166380109329,

%T 417542026135897,417542026135897,13805828672331787

%N Odd numbers x such that x and x + 2 are both sums of divisors, i.e., elements of A000203.

%C If some x or x + 2 is in A300869, i.e., it has more than one representation as sigma(m), as for x = 417542026135897 = sigma((4*17*209459)^2) = sigma((5*17*209459)^2) = sigma((2*7723267)^2) - 2, then it is listed with multiplicity and all corresponding pairs of numbers are provided in A300780.

%H Seqfan Mailing List Thread, <a href="http://list.seqfan.eu/oldermail/seqfan/2018-March/018439.html">Consecutive sigmas</a>, with contributions from _Franklin T. Adams-Watters_, _Hugo Pfoertner_ and _Emmanuel Vantieghem_.

%e a(1) = 1 because 1 = sigma(1) and 3 = sigma(2),

%e a(2) = 13: 13 = sigma(9) and 15 = sigma(8),

%e a(3) = 91: 91 =sigma(36), 93 = sigma(50),

%e a(4) = 241573: 241573 = sigma(241081), 241575 = sigma(117128),

%e a(5) = 38152387: 38152387 = sigma(15069924), 38152389 = sigma(23011209).

%Y Cf. A000203, A002191, A083531, A300780 (numbers corresponding to sigma values), A300869.

%K nonn,hard,more

%O 1,2

%A _Hugo Pfoertner_, Mar 12 2018

%E a(5) from _Emmanuel Vantieghem_

%E a(6)-a(11) from _Giovanni Resta_, Mar 13 2018