cp's OEIS Frontend

A277794 Numbers k such that the sum of proper divisors of k is a prime, and the sum of the numbers less than k that do not divide k is also a prime.

%I A277794 #23 Jan 16 2023 04:48:55
%S A277794 4,21,85,129,201,237,324,325,517,549,669,837,865,1081,1137,1161,1165,
%T A277794 1309,1389,1677,1765,2169,2233,2304,2305,2469,2709,2737,2761,3265,
%U A277794 3297,3745,3961,4161,4285,4693,4705,4741,4989,5061,5221,5349,5817,5949,6249,6457,6517,6685,6789,6813,6853,6921
%N A277794 Numbers k such that the sum of proper divisors of k is a prime, and the sum of the numbers less than k that do not divide k is also a prime.
%C A277794 Intersection of A037020 and A200981.
%C A277794 Numbers k such that A000005(A001065(k)) = A000005(A024816(k)) = 2 or A000005(A000203(k) - k) = A000005(A000217(k) - A000203(k)) = 2.
%C A277794 All terms are composite (A002808).
%H A277794 Robert Israel, <a href="/A277794/b277794.txt">Table of n, a(n) for n = 1..1000</a>
%H A277794 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>
%e A277794 21 is in the sequence because 21 has three proper divisors {1, 3, 7}, and therefore seventeen non-divisors {2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, so the sum of proper divisors is 1 + 3 + 7 = 11 (which is prime) and the sum of non-divisors is 2 + 4 + 5 + 6 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 = 199 (which is also prime).
%e A277794 22 is not in the sequence because its three proper divisors {1, 2, 11} add up to 14, which is composite.
%p A277794 f:= proc(n) local t; t:= numtheory:-sigma(n) - n; isprime(t) and isprime(n*(n-1)/2 - t) end proc:
%p A277794 select(f, [$1..10^4]); # _Robert Israel_, Nov 10 2016
%t A277794 Select[Range[7000], DivisorSigma[0, #1 ((#1 + 1)/2) - DivisorSigma[1, #1]] == 2 && DivisorSigma[0, DivisorSigma[1, #1] - #1] == 2 & ]
%Y A277794 Cf. A000005, A000203, A000217, A002808, A037020, A200981.
%K A277794 nonn,easy
%O A277794 1,1
%A A277794 _Ilya Gutkovskiy_, Oct 31 2016