A011264 -id:A011264 - cp's OEIS frontend

A011262 In the prime factorization of n, increment odd powers and decrement even powers (multiplicative and self-inverse).

1, 4, 9, 2, 25, 36, 49, 16, 3, 100, 121, 18, 169, 196, 225, 8, 289, 12, 361, 50, 441, 484, 529, 144, 5, 676, 81, 98, 841, 900, 961, 64, 1089, 1156, 1225, 6, 1369, 1444, 1521, 400, 1681, 1764, 1849, 242, 75, 2116, 2209, 72, 7, 20, 2601, 338, 2809, 324, 3025, 784, 3249

Offset: 1

Marc LeBrun

Cf. A001221, A011264, A027748, A103889, A124010.

a011262 n = product $ zipWith (^)
                      (a027748_row n) (map a103889 $ a124010_row n)
-- Reinhard Zumkeller, Jun 23 2013

f[n_, k_] := n^(If[EvenQ[k], k - 1, k + 1]); Table[Times @@ f @@@ FactorInteger[n], {n, 57}] (* Jayanta Basu, Aug 14 2013 *)

a(n)=my(f=factor(n));return(prod(i=1,#f[,1],f[i,1]^(f[i,2]-(-1)^f[i,2]))) \\ Paul Tek, Jun 01 2013

Multiplicative with f(p^k) = p^(k-1) if k even, p^(k+1) if k odd.
a(n) = Product_{k = 1..A001221(n)} A027748(n,k) ^ A103889(A124010(n,k)). - Reinhard Zumkeller, Jun 23 2013
Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} ((p^5 + p^4 - p + 1)/(p^5 + p^4 + p^3 + p^2)) = 0.21311151701724196530... . - Amiram Eldar, Oct 13 2022

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

Original entry on oeis.org

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

Alex Ratushnyak, Jul 07 2013

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

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

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

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

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

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 *)

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

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=', ')

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

Keyword:mult added by Andrew Howroyd, Jul 23 2018

A366423 Multiplicative with a(p^e) = p^(e+1-p) if p|e, and p^(e+1) otherwise.

Original entry on oeis.org

1, 4, 9, 2, 25, 36, 49, 16, 27, 100, 121, 18, 169, 196, 225, 8, 289, 108, 361, 50, 441, 484, 529, 144, 125, 676, 3, 98, 841, 900, 961, 64, 1089, 1156, 1225, 54, 1369, 1444, 1521, 400, 1681, 1764, 1849, 242, 675, 2116, 2209, 72, 343, 500, 2601, 338, 2809, 12, 3025

Offset: 1

Amiram Eldar, Nov 17 2023

A permutation of the positive integers. 1 is the only fixed point.

a(n) is a powerful number (A001694) if and only if n is not in A100717.

Cf. A000040, A001694, A005117, A007947, A048102, A072873, A100717, A103889, A130508, A342090.

Similar sequences: A011262, A011264.

f[p_, e_] := p^(e + 1 + If[Mod[e, p] == 0, -p, 0]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(f[i,2] + 1 + if(!(f[i,2]%f[i,1]), -f[i,1])));}

a(2^e) = 2^A103889(e).
a(3^e) = 3^A130508(e).
A007947(a(n)) = A007947(n).
a(A051674(n)) = A000040(n).
a(n) is squarefree (A005117) if and only if n is in A048102.
a(A048102(n)) = A007947(A048102(n)).
a(n) == 0 (mod n) if and only if n is not in A342090.
a(n) | n if and only if n is in A072873.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Product_{p prime} (1 - 1/p + 1/(1 + p) - (p-1)/(p^p * (1 + p^p))) = 0.660264348361... .

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A011262 In the prime factorization of n, increment odd powers and decrement even powers (multiplicative and self-inverse).

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

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A366423 Multiplicative with a(p^e) = p^(e+1-p) if p|e, and p^(e+1) otherwise.

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