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A248101 Completely multiplicative with a(prime(n)) = prime(n)^((n+1) mod 2). 5
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 (list; graph; refs; listen; history; text; internal format)
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: A161788 A170819 A140211 * A097706 A132740 A106621

Adjacent sequences:  A248098 A248099 A248100 * A248102 A248103 A248104

KEYWORD

nonn,mult

AUTHOR

Tom Edgar, Mar 03 2015

STATUS

approved

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Last modified May 7 10:15 EDT 2021. Contains 343649 sequences. (Running on oeis4.)