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A081084 Nonsquarefree numbers m such that rad(m+1)=rad(m)+1, where rad(m)=A007947(m) is the squarefree kernel of m. 2

%I #18 Sep 29 2018 10:19:05

%S 8,48,224,960,65024,261120,1046528,4190208,268402688,1073676288,

%T 4294836224,17179607040,70368727400448,4503599493152768,

%U 18014398241046528,72057593501057024,288230375077969920

%N Nonsquarefree numbers m such that rad(m+1)=rad(m)+1, where rad(m)=A007947(m) is the squarefree kernel of m.

%C For k >= 3, 2^k*(2^(k-2)-1) is in the sequence if and only if 2^(k-1)-1 and 2^(k-2)-1 are squarefree. So if m is a term, m+1=2^(k-1)-1 is a squarefree number squared. - Lambert Herrgesell (zero815(AT)googlemail.com), Feb 18 2007

%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 48, p. 18, Ellipses, Paris 2008.

%e 48 = 2^4*3 is in the sequence because it is not squarefree, its squarefree kernel is 6 and the squarefree kernel of 49 = 7^2 is 7.

%p with(numtheory): rad:=proc(n) local fs, c: fs:=convert(factorset(n),list): c:=nops(fs): product(fs[j],j=1..c) end: b:=proc(n) if mobius(n)=0 and rad(n+1)=rad(n)+1 then n else fi end:seq(b(n),n=1..1000); # _Emeric Deutsch_

%o (PARI) rad(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])

%o is(n)=!issquarefree(n) && rad(n+1)==rad(n)+1 \\ _Charles R Greathouse IV_, Aug 08 2013

%Y Cf. A081083, A062503.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, Mar 04 2003

%E a(5)-a(8) from _Emeric Deutsch_, Mar 29 2005

%E Edited and a(9) onwards supplied by Lambert Herrgesell (zero815(AT)googlemail.com), Feb 18 2007

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