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A210445 Least positive integer k with k*n practical. 3

%I

%S 1,1,2,1,4,1,4,1,2,2,6,1,6,2,2,1,12,1,12,1,2,3,12,1,4,3,2,1,12,1,16,1,

%T 2,6,4,1,18,6,2,1,20,1,20,2,2,6,24,1,4,2,4,2,24,1,4,1,4,6,24,1,24,8,2,

%U 1,4,1,30,3,4,2,30,1,30,9,2,3,4,1,36,1,2,10,36,1,4,10,4,1,36,1,4,3,6,12,4,1,42,2,2,1

%N Least positive integer k with k*n practical.

%C Conjecture: a(n) < n for all n>1, and a(n) < n/2 for all n>47.

%C Large values are obtained for prime n: The corresponding subsequence is a(p(n)) = (1, 2, 4, 4, 6, 6, 12, 12, 12, 12, 16, 18, 20, 20, 24, 24, 24, 24, ...), while for composite indices, a(c(n)) = (1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 3, 1, 4, 3, 2, 1, 1, 1, 2, ...). - _M. F. Hasler_, Jan 21 2013

%H Zhi-Wei Sun, <a href="/A210445/b210445.txt">Table of n, a(n) for n = 1..10000</a>

%H G. Melfi, <a href="http://dx.doi.org/10.1006/jnth.1996.0012">On two conjectures about practical numbers</a>, J. Number Theory 56 (1996) 205-210 [<a href="http://www.ams.org/mathscinet-getitem?mr=1370203">MR96i:11106</a>].

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588 [math.NT], 2012-2015.

%F a(n) = 1 iff n is in A005153, therefore a(n) > 1 for all odd n>1. - _M. F. Hasler_, Jan 21 2013

%e a(10)=2 since 2*10=20 is practical but 1*10=10 is not.

%t f[n_]:=f[n]=FactorInteger[n]

%t Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])

%t Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}]

%t pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)

%t Do[Do[If[pr[k*n]==True,Print[n," ",k];Goto[aa]],{k,1,n}];

%t Print[n," ",counterexample];Label[aa];Continue,{n,1,100}]

%o (PARI) A210445(n)={for(k=1,n,is_A005153(k*n)&&return(k))} \\ (Would return 0 if a(n)>n.) - _M. F. Hasler_, Jan 20 2013

%Y Cf. A005153, A210444, A071558, A208243, A208244, A208246, A208249, A209253, A209254, A209312, A219185, A219312, A219315, A219320.

%K nonn,nice

%O 1,3

%A _Zhi-Wei Sun_, Jan 20 2013

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Last modified October 16 06:30 EDT 2019. Contains 328050 sequences. (Running on oeis4.)