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A247503
Completely multiplicative with a(prime(n)) = prime(n)^(n mod 2).
8
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.
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
First 28 terms are the same as A343430.
Sequence in context: A352458 A132741 A072436 * A343430 A292895 A371015
KEYWORD
nonn,mult
AUTHOR
Tom Edgar, Mar 03 2015
STATUS
approved