A355419 a(n) is the number of solutions to x^y == y^x (mod p) where 0 < x,y <= p and p is the n-th prime.

2, 3, 7, 17, 21, 29, 39, 59, 79, 77, 101, 101, 107, 117, 161, 183, 177, 183, 205, 239, 293, 253, 241, 359, 339, 343, 337, 319, 347, 421, 411, 403, 471, 435, 467, 483, 581, 527, 535, 589, 549, 651, 715, 703, 661, 673, 763, 765, 707, 833, 819, 793, 1009, 829

Offset

1,1

Links

Table of n, a(n) for n=1..54.

Formula

a(n) = A355486(n) + A000040(n).

a(n) = p+2*(number of solutions to x^y == y^x (mod p) where 1 < x < y < p). - Chai Wah Wu, Aug 30 2022

Example

Solutions for a(3):
  p  x  y  x^y mod p  y^x mod p
  -  -  -  ---------  ---------
  5  1  1      1          1
  5  2  2      4          4
  5  2  4      1          1
  5  3  3      2          2
  5  4  2      1          1
  5  4  4      1          1
  5  5  5      0          0
Total number of solutions: 7.

Prog

(Python)
from sympy import prime
def f(n):
  S = 0
  for x in range(1, n + 1):
    for y in range(x + 1 , n + 1):
      if ((pow(x, y, n) == pow(y, x, n))):
        S += 2
  return S + n
def a(n): return f(prime(n))
(Python)
from sympy import prime
def A355419(n): return (p:=prime(n))+sum(2 for x in range(2, p-1) for y in range(x+1, p) if pow(x, y, p)==pow(y, x, p)) # Chai Wah Wu, Aug 30 2022
(PARI) a(n) = my(p=prime(n)); sum(x=1, p, sum(y=1, p, Mod(x, p)^y == Mod(y, p)^x)); \\ Michel Marcus, Jul 05 2022

Crossrefs

Cf. A000040, A355486, A355069.

Sequence in context: A256917 A089144 A171430 * A045332 A045333 A040141

Adjacent sequences: A355416 A355417 A355418 * A355420 A355421 A355422

Keyword

nonn

Author

Darío Clavijo, Jul 01 2022

Status

approved