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

1,3

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.

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+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

Example

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A351518 A170819 A140211 * A097706 A132740 A106621

Adjacent sequences: A248098 A248099 A248100 * A248102 A248103 A248104

Keyword

nonn,mult

Author

Tom Edgar, Mar 03 2015

Status

approved