A235139 Twin primes p, p+2 such that p+1 is a primary pseudoperfect number.

5, 7, 41, 43, 47057, 47059

Offset: 1

Author

Jonathan Sondow and Emmanuel Tsukerman, Jan 04 2014

Keywords

• more
• hard
• nonn

Comments

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.

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.

See A235364 for a similar property of Giuga numbers.

Examples

			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.
		

Links

• J. Sondow and E. Tsukerman, The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers, arXiv:1401.0322 [math.NT], 2014; see section 4.
• MathWorld, Giuga Number
• Wikipedia, Giuga number
• Wikipedia, Primary pseudoperfect number

Crossrefs

Cf. A007850, A054377.

Programs

Mathematica

A054377 = Cases[Import["https://oeis.org/A054377/b054377.txt", "Table"], {, }][[All, 2]];
lst = {}; For[i = 1, i <= Length[A054377], i++, n = A054377[[i]];
If[PrimeQ[n + 1] && PrimeQ[n - 1], AppendTo[lst, n - 1]; AppendTo[lst, n + 1]]]; lst (* Robert Price, Mar 14 2020 *)