A248101 - OEIS

%I #35 Apr 06 2017 02:34:17

%S 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,

%T 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,

%U 61,1,63,1,13,3,1,1,3,7,71,9,1,37,3,19,7,39,79

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

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

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

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

%H Reinhard Zumkeller, <a href="/A248101/b248101.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>

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

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

%F a(n) = Product(A027746(n,k): k = 1 .. A001222(n) and A049084(A027746(n,k)) is even). - _Reinhard Zumkeller_, Mar 06 2015

%e Since 10 = 2*5, 2 = prime(1), and 5 = prime(3), a(10) = 1*1 = 1.

%e Since 9 = 3^2 and 3 is an even-indexed prime, 3 = prime(2), then a(9) = 3^2 = 9.

%e Since 35 = 5*7, 5 = prime(3), and 7 = prime(4), we see that a(35) = 1*7 = 7.

%t 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 *)

%t Array[Times @@ (FactorInteger[#] /. {p_, e_} /; e > 0 :> (p^Mod[PrimePi@ p + 1, 2])^e) &, 79] (* _Michael De Vlieger_, Apr 05 2017 *)

%o (Sage)

%o n=100; evenIndexPrimes=[primes_first_n(2*n+2)[2*i+1] for i in [0..n]]

%o [prod([(x[0]^(x[0] in evenIndexPrimes))^x[1] for x in factor(n)]) for n in [1..n]]

%o (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

%o (Haskell)

%o a248101 = product . filter (even . a049084) . a027746_row

%o -- _Reinhard Zumkeller_, Mar 06 2015

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

%K nonn,mult

%O 1,3

%A _Tom Edgar_, Mar 03 2015