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

1,2

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.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Index to divisibility sequences

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

Example

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.

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 *)

Prog

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

Crossrefs

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

First 28 terms are the same as A343430.

Sequence in context: A352458 A132741 A072436 * A343430 A292895 A371015

Adjacent sequences: A247500 A247501 A247502 * A247504 A247505 A247506

Keyword

nonn,mult

Author

Tom Edgar, Mar 03 2015

Status

approved