A247503 Completely multiplicative with a(prime(n)) = prime(n)^(n mod 2).

1, 2, 1, 4, 5, 2, 1, 8, 1, 10, 11, 4, 1, 2, 5, 16, 17, 2, 1, 20, 1, 22, 23, 8, 25, 2, 1, 4, 1, 10, 31, 32, 11, 34, 5, 4, 1, 2, 1, 40, 41, 2, 1, 44, 5, 46, 47, 16, 1, 50, 17, 4, 1, 2, 55, 8, 1, 2, 59, 20, 1, 62, 1, 64, 5, 22, 67, 68, 23, 10, 1, 8, 73, 2, 25, 4

Offset: 1

Author

Tom Edgar, Mar 03 2015

Keywords

• nonn
• mult

Comments

To compute a(n) replace even-indexed primes in the prime factorization of n by 1.

a(p) = p if p is in A031368.

a(p) = 1 if p is in A031215.

Examples

			Since 10 = 2*5, 2 = prime(1), and 5 = prime(3), a(10) = 2*5 = 10.
Since 9 = 3^2 and 3 is an even-indexed prime, 3 = prime(2), then a(9) = 1^2 = 1.
Since 30 = 2*3*5, 2 = prime(1), 3 = prime(2), and 5 = prime(3), we see that a(30) = 2*1*5 = 10.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Index to divisibility sequences

Crossrefs

Cf. A031368, A031215, A248101, A001222, A027746, A049084.

First 28 terms are the same as A343430.

Programs

Haskell

a247503 = product . filter (odd . a049084) . a027746_row
-- Reinhard Zumkeller, Mar 06 2015

Mathematica

f[n_] := Block[{a, g, pf = FactorInteger@ n}, a = PrimePi[First /@ pf]; g[x_] := If[EvenQ@ x, 1, Prime@ x]; Times @@ Power @@@ Transpose@ {g /@ a, Last /@ pf}]; Array[f, 120] (* Michael De Vlieger, Mar 03 2015 *)
Array[Times @@ (FactorInteger[#] /. {p_, e_} /; e > 0 :> (p^Mod[PrimePi@ p, 2])^e) &, 76] (* Michael De Vlieger, Apr 05 2017 *)

PARI

a(n) = {f = factor(n); for (i=1, #f~, f[i,2] *= (primepi(f[i,1]) % 2);); factorback(f);} \\ Michel Marcus, Mar 03 2015

Python

from math import prod
from sympy import factorint, primepi
def A247503(n): return prod(p**e for p, e in factorint(n).items() if primepi(p)&1) # Chai Wah Wu, Dec 26 2022

Sage

n=100; oddIndexPrimes=[primes_first_n(2*n+1)[2*i] for i in [0..n]]
[prod([(x[0]^(x[0] in oddIndexPrimes))^x[1] for x in factor(n)]) for n in [1..n]]

Formula

When n = Product_{k>=1} prime(k)^r_k, a(n) = Product_{k>=1} prime(k)^(r_k*(k mod 2)).

a(n) = n/A248101(n).

a(n) = Product_{k = 1..A001222(n)} A027746(n,k) and A049084(A027746(n,k)) is odd). - Reinhard Zumkeller, Mar 06 2015