[seqfan] Re: mth squarefree = mth prime power

[Richard Mathar]

lq> Leroy Quet q1qq2qqq3qqqq at yahoo.com
lq> Sat May 16 18:45:48 CEST 2009
lq> 
lq> I just sent in this sequence. Is it complete?
lq> 
lq> %I A160513
lq> %S A160513 1,2,3,7,17,19
lq> %N A160513 Positive squarefree integers n such that, if n is the mth squarefree integer, then n is the mth power of a prime as well. 
lq> %C A160513 Is this sequence complete? 
lq> %C A160513 1 here is considered to be both the first squarefree integer and the first power of a prime. 
lq> %Y A160513 A000961,A005117 
lq> %K A160513 fini,more,nonn
lq> %O A160513 1,2
lq> 
lq> 
lq> Since the squarefree integers have a density of 6/pi^2; and since the density, as n approaches infinity, of the number of prime powers <= n approaches 0 (right?), then it is certain that this sequence is finite.
lq> (Unless I am missing something here.)

An exhaustive search shows that in the index range m <= 100000, these 6 numbers
are the only A000961(m) such that A000961(m)=A005117(m); that is, the
next entry would have to be larger than 1295947. So it is very likely
that the sequence contains only these 6 numbers.

In Maple:

A000961 := proc(n)
	option remember;
	if n = 1 then
		1;
	else
		for a from procname(n-1)+1 do
			if nops( numtheory[factorset](a)) = 1 then
				RETURN(a) ;
			fi;
		od:
	fi;
end:
A005117 := proc(n)
	option remember;
	if n = 1 then
		1;
	else
		for a from procname(n-1)+1 do
			if numtheory[issqrfree](a) then
				RETURN(a) ;
			fi;
		od:
	fi;
end:
A160513m := proc(n)
	option remember;
	if n = 1 then
		1;
	else
		for a from procname(n-1)+1 do
			if A000961(a) = A005117(a) then
				RETURN(a) ;
			fi;
		od:
	fi;
end:
for n from 1 do
	print(n,A000961(A160513m(n))) ;
od: