A227324 Result of changing both the prime indices and the exponents in the prime factorization of n: increment odd values, decrement even values.

1, 9, 4, 3, 49, 36, 25, 81, 2, 441, 169, 12, 121, 225, 196, 27, 361, 18, 289, 147, 100, 1521, 841, 324, 7, 1089, 16, 75, 529, 1764, 1369, 729, 676, 3249, 1225, 6, 961, 2601, 484, 3969, 1849, 900, 1681, 507, 98, 7569, 2809, 108, 5, 63, 1444, 363, 2209, 144

Offset: 1

Author

Alex Ratushnyak, Jul 07 2013

Keywords

• nonn
• easy
• mult

Comments

A self-inverse permutation on the positive integers: a(a(n)) = n.

Examples

			n = 2^3 => a(n) = 3^4 = 81.
n = 3^2 => a(n) = 2^1 = 2.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003958-A003965, A011262, A011264, A045965-A045973, A151799, A151800.

Programs

Maple

a:= n-> mul(ithprime(i[1])^i[2], i=map(x->map(y->y-1+2*irem(y, 2),
        [numtheory[pi](x[1]), x[2]]), ifactors(n)[2])):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 17 2013

Mathematica

a[n_] := If[n == 1, 1, Product[{p, e} = pe; Prime[BitXor[PrimePi[p] - 1, 1] + 1]^(BitXor[e - 1, 1] + 1), {pe, FactorInteger[n]}]];
Array[a, 100] (* Jean-François Alcover, May 31 2019, after Andrew Howroyd *)

PARI

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); prime( bitxor( primepi(p)-1, 1)+1)^(bitxor(e-1, 1)+1))} \\ Andrew Howroyd, Jul 23 2018

Python

primes = [2]*2
primes[1] = 3
def addPrime(k):
  for p in primes:
    if k%p==0:  return
    if p*p > k:  break
  primes.append(k)
for n in range(5, 1000000, 6):
  addPrime(n)
  addPrime(n+2)
for n in range(1,99):
  p = 1
  j = n
  i = 0
  while j>1:
    e = 0
    while j % primes[i] == 0:
      j /= primes[i]
      e+=1
    if e:
      e = ((e-1)^1) + 1
      p*= primes[i^1]**e
    i += 1
  print(str(p), end=', ')

Formula

Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} ((p-1)*(p^6 + q(p) +(p^3-1)*q(p)^2))/(p^7 - p*q(p)^2) = 0.3120270364..., where q(p) = nextprime(p) = A151800(p) if p has an odd index, and q(p) = prevprime(p) = A151799(p) otherwise. - Amiram Eldar, Sep 17 2023

Extensions

Keyword:mult added by Andrew Howroyd, Jul 23 2018