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

8, 48, 224, 960, 65024, 261120, 1046528, 4190208, 268402688, 1073676288, 4294836224, 17179607040, 70368727400448, 4503599493152768, 18014398241046528, 72057593501057024, 288230375077969920

Offset: 1

Author

Reinhard Zumkeller, Mar 04 2003

Keywords

• nonn

Comments

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

Examples

			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.
		

References

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

Crossrefs

Cf. A081083, A062503.

Programs

Maple

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

PARI

rad(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])
is(n)=!issquarefree(n) && rad(n+1)==rad(n)+1 \\ Charles R Greathouse IV, Aug 08 2013

Extensions

a(5)-a(8) from Emeric Deutsch, Mar 29 2005

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