%I #20 Mar 14 2020 18:53:26
%S 5,7,41,43,47057,47059
%N Twin primes p, p+2 such that p+1 is a primary pseudoperfect number.
%C Same as twin primes p, p+2 such that (p+1)*(p+2) is a primary pseudoperfect number (A054377). Appears also to be same as twin primes p, p+2 such that p*(p+1) is a Giuga number (A007850). See the link "The p-adic order of power sums...": Theorem 8, Example 1, and Question 1.
%C Is it always true that if a primary pseudoperfect number N > 2 is adjacent to a prime N-1 or N+1, then in fact N lies between twin primes N-1, N+1? For all 7 known primary pseudoperfect numbers N > 2, either both N-1 and N+1 are prime or neither is prime.
%C See A235364 for a similar property of Giuga numbers.
%H J. Sondow and E. Tsukerman, <a href="https://arxiv.org/abs/1401.0322">The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers</a>, arXiv:1401.0322 [math.NT], 2014; see section 4.
%H MathWorld, <a href="http://mathworld.wolfram.com/GiugaNumber.html">Giuga Number</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Giuga_number">Giuga number</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Primary_pseudoperfect_number">Primary pseudoperfect number</a>
%e For the twin primes (p,p+2) = (5, 7), (41, 43), (47057, 47059), the numbers p+1 = 6, 42, 47058 and (p+1)*(p+2) = 42, 1806, 2214502422 are primary pseudoperfect numbers, and p*(p+1) = 30, 1722, 2214408306 are Giuga numbers.
%t A054377 = Cases[Import["https://oeis.org/A054377/b054377.txt", "Table"], {_, _}][[All, 2]];
%t lst = {}; For[i = 1, i <= Length[A054377], i++, n = A054377[[i]];
%t If[PrimeQ[n + 1] && PrimeQ[n - 1], AppendTo[lst, n - 1]; AppendTo[lst, n + 1]]]; lst (* _Robert Price_, Mar 14 2020 *)
%Y Cf. A007850, A054377.
%K more,hard,nonn
%O 1,1
%A _Jonathan Sondow_ and Emmanuel Tsukerman, Jan 04 2014