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A076734 Smallest squarefree number greater than or equal to n having the same number of prime factors as n (counted with multiplicity). 5
1, 2, 3, 6, 5, 6, 7, 30, 10, 10, 11, 30, 13, 14, 15, 210, 17, 30, 19, 30, 21, 22, 23, 210, 26, 26, 30, 30, 29, 30, 31, 2310, 33, 34, 35, 210, 37, 38, 39, 210, 41, 42, 43, 66, 66, 46, 47, 2310, 51, 66, 51, 66, 53, 210, 55, 210, 57, 58, 59, 210, 61, 62, 66, 30030, 65, 66, 67 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) = n if and only if n is squarefree; A001222(a(n)) = A001222(n).

Suppose k = the number of prime factors of n. If p_k# is the product of the first k primes (i.e., a primorial), then the squarefree number a(n) will be p_k# if and only if p_k# <= n. This is because the smallest squarefree number with k prime factors is p_k#. - Michael De Vlieger, Aug 31 2014

LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..10000

EXAMPLE

a(7) = 7 because 7 is squarefree.

a(8) = 30 because 8 has 3 prime factors but is not squarefree; 12, 18, 20 and 27 also have 3 prime factors each but are not squarefree either; so 30 is the smallest squarefree number with 3 prime factors.

a(9) = 10 because 9 has 2 prime factors but is not squarefree, while 10 has 2 prime factors and is squarefree.

MAPLE

f:= proc(n)

     uses numtheory, Optimization;

     local k, P, m, Q;

     if issqrfree(n) then return n fi;

     k:= bigomega(n);

     m:= floor((n-1)/2);

     P:= select(isprime, {2, seq(2*i+1, i=1..m)});

     while nops(P) < k do

        m:= m+1;

        if isprime(2*m+1) then P:= P union {2*m+1} fi

     od:

     if convert(P[1..k], `*`) > n then return convert(P[1..k], `*`) fi;

     Q:= Minimize(add(x[i]*log(P[i]), i=1..nops(P)),

         { add(x[i]*log(P[i]), i=1..nops(P)) >= log(n),

           add(x[i], i=1..nops(P))=k}, assume=binary);

     simplify(exp(Q[1]));

end proc:

seq(f(n), n=1..100); # Robert Israel, Sep 01 2014

MATHEMATICA

f[n_, lim_] := If[n == 0, {1}, Block[{P = Product[Prime@ i, {i, n}], k = 1, c, w = ConstantArray[1, n]}, {P}~Join~Reap[Do[w = If[k == 1, MapAt[# + 1 &, w, -k], Join[Drop[MapAt[# + 1 &, w, -k], -k + 1], ConstantArray[1, k - 1]]]; c = Times @@ Map[If[# == 0, 1, Prime@#] &, Accumulate@ w]; If[c < lim, Sow[c]; k = 1, If[k == n, Break[], k++]], {i, Infinity}]][[-1, 1]]]]; Array[Which[SquareFreeQ@ #1, #1, #3 < #1, #3, True, SelectFirst[Sort@ f[#2, #1 + Product[Prime@ i, {i, 1 + #2}]], Function[k, k > #1]]] & @@ {#, PrimeOmega@ #, Times @@ Prime@ Range@ #} &, 10^4] (* Michael De Vlieger, Oct 20 2017 *)

CROSSREFS

Cf. A002110, A005117, A067535.

Sequence in context: A335835 A142151 A003968 * A242314 A242311 A097723

Adjacent sequences:  A076731 A076732 A076733 * A076735 A076736 A076737

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Nov 08 2002

STATUS

approved

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Last modified September 19 17:12 EDT 2020. Contains 337178 sequences. (Running on oeis4.)