login
Number of divisors of n which are not greater than the cubefree kernel of n.
5

%I #15 Jul 18 2017 11:43:24

%S 1,2,2,3,2,4,2,3,3,4,2,6,2,4,4,3,2,6,2,6,4,4,2,7,3,4,3,6,2,8,2,3,4,4,

%T 4,9,2,4,4,7,2,8,2,6,6,4,2,7,3,6,4,6,2,6,4,7,4,4,2,12,2,4,6,3,4,8,2,6,

%U 4,8,2,11,2,4,6,6,4,8,2,8,3,4,2,12,4,4,4,7,2,12,4,6,4,4,4,7,2,6,6,9,2,8

%N Number of divisors of n which are not greater than the cubefree kernel of n.

%C a(n) >= A073184(n).

%H Antti Karttunen, <a href="/A073182/b073182.txt">Table of n, a(n) for n = 1..10000</a>

%e The cubefree kernel of 56 = 7*2^3 is 28 = 7*2^2 and the divisors <= 28 of 56 are {1, 2, 4, 7, 8, 14, 28}, therefore a(56) = 7.

%t Table[Function[k, DivisorSum[n, 1 &, # <= k &]]@ Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 0 :> p^Min[e, 2]], {n, 102}] (* _Michael De Vlieger_, Jul 18 2017 *)

%o (PARI) a007948(n) = my(f=factor(n)); for (i=1, #f~, f[i, 2] = min(f[i, 2], 2)); factorback(f);

%o a(n) = my(cfk = a007948(n)); sumdiv(n, d, d<=cfk); \\ _Michel Marcus_, May 14 2015

%Y Cf. A000005, A007948, A073183, A073180.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Jul 19 2002