A235364 -id:A235364 - cp's OEIS frontend

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

5, 7, 41, 43, 47057, 47059

Offset: 1

Jonathan Sondow and Emmanuel Tsukerman, Jan 04 2014

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.

			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.

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

Cf. A007850, A054377.

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 *)

A235363 (1 + Sum_{k=1..m-1} k^(m-1)) (mod m), for m = 1, 3, 5, 7, 9, ...

Original entry on oeis.org

0, 0, 0, 0, 7, 0, 0, 11, 0, 0, 15, 0, 21, 19, 0, 0, 23, 1, 0, 27, 0, 0, 22, 0, 43, 35, 0, 1, 39, 0, 0, 43, 53, 0, 47, 0, 0, 51, 1, 0, 55, 0, 69, 59, 0, 79, 63, 1, 0, 67, 0, 0, 50, 0, 0, 75, 0, 1, 79, 1, 111, 83, 101, 0, 87, 0, 115, 91, 0, 0, 95, 1, 117, 99, 0, 0, 103, 1, 0, 107, 1, 0, 78, 0, 157, 115, 0, 151, 119, 0, 0, 123, 149, 1, 127, 0, 0, 131, 0, 0, 135

Offset: 0

Jonathan Sondow, Jan 07 2014

nonn

a(n) = (1 + Sum_{k=1..2*n} k^(2*n)) (mod 2*n+1), for n = 0, 1, 2, 3, ...

The Agoh-Giuga Conjecture is that a(n)=0 iff 2*n+1 is 1 or a prime.

MathWorld, Giuga's Conjecture
Wikipedia, Agoh-Giuga conjecture

Cf. A007850, A046094, A204187, A228037, A235364.

Table[ Mod[ Sum[ PowerMod[ k, n - 1, n], {k, n - 1}] + 1, n], {n, 1, 201, 2}]

a(n) = 0 iff A235364(n) = 0.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica