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%I #8 May 30 2020 04:07:24
%S 672,523776,19327369215
%N Numbers k such that the abundance (A033880) of k is equal to the deficiency (A033879) of k+1.
%C Equivalently, k and k+1 have the same absolute value of abundance (or deficiency) with opposite signs.
%C Equivalently, s(k) + s(k+1) = k + (k+1), where s(k) is the sum of proper divisors of k (A001065).
%C If k is a 3-perfect number (A005820) and k+1 is a prime, then k is in the sequence. Of the 6 known 3-perfect numbers only 672 and 523776 have this property.
%C a(4) > 10^11, if it exists.
%C a(4) > 10^13, if it exists. - _Giovanni Resta_, May 30 2020
%e 672 is a term since A033880(672) = sigma(672) - 2*672 = 2016 - 1344 = 672, and A033879(673) = 2*673 - sigma(673) = 1346 - 674 = 672.
%t ab[n_] := DivisorSigma[1, n] - 2*n; Select[Range[6 * 10^5], ab[#] == -ab[# + 1] &]
%Y Cf. A000203, A001065, A005820, A033879, A033880, A330901, A335253.
%K nonn,hard,bref,more
%O 1,1
%A _Amiram Eldar_, May 28 2020
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