A248101 Completely multiplicative with a(prime(n)) = prime(n)^((n+1) mod 2).

1, 1, 3, 1, 1, 3, 7, 1, 9, 1, 1, 3, 13, 7, 3, 1, 1, 9, 19, 1, 21, 1, 1, 3, 1, 13, 27, 7, 29, 3, 1, 1, 3, 1, 7, 9, 37, 19, 39, 1, 1, 21, 43, 1, 9, 1, 1, 3, 49, 1, 3, 13, 53, 27, 1, 7, 57, 29, 1, 3, 61, 1, 63, 1, 13, 3, 1, 1, 3, 7, 71, 9, 1, 37, 3, 19, 7, 39, 79

Offset: 1

Author

Tom Edgar, Mar 03 2015

Keywords

• nonn
• mult

Comments

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

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

a(p) = 1 if p is in A031368.

Examples

			Since 10 = 2*5, 2 = prime(1), and 5 = prime(3), a(10) = 1*1 = 1.
Since 9 = 3^2 and 3 is an even-indexed prime, 3 = prime(2), then a(9) = 3^2 = 9.
Since 35 = 5*7, 5 = prime(3), and 7 = prime(4), we see that a(35) = 1*7 = 7.
		

Links

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

Crossrefs

Cf. A031368, A031215, A247503, A001222, A027746, A049084.

Programs

Haskell

a248101 = product . filter (even . a049084) . a027746_row
-- Reinhard Zumkeller, Mar 06 2015

Mathematica

f[n_] := Block[{a, g, pf = FactorInteger@ n}, a = PrimePi[First /@ pf]; g[x_] := If[Or[OddQ@ x, x == 0], 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 + 1, 2])^e) &, 79] (* Michael De Vlieger, Apr 05 2017 *)

PARI

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

Sage

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

a(n) = n / A247503(n).

a(n) = Product(A027746(n,k): k = 1 .. A001222(n) and A049084(A027746(n,k)) is even). - Reinhard Zumkeller, Mar 06 2015