|
|
A210445
|
|
Least positive integer k with k*n practical.
|
|
4
|
|
|
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, 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, 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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
Conjecture: a(n) < n for all n>1, and a(n) < n/2 for all n>47.
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
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
a(10)=2 since 2*10=20 is practical but 1*10=10 is not.
|
|
MATHEMATICA
|
f[n_]:=f[n]=FactorInteger[n]
Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2])
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}]
pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0)
Do[Do[If[pr[k*n]==True, Print[n, " ", k]; Goto[aa]], {k, 1, n}];
Print[n, " ", counterexample]; Label[aa]; Continue, {n, 1, 100}]
|
|
PROG
|
|
|
CROSSREFS
|
Cf. A005153, A210444, A071558, A208243, A208244, A208246, A208249, A209253, A209254, A209312, A219185, A219312, A219315, A219320.
|
|
KEYWORD
|
nonn,nice
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|