login
Numbers k such that k and k+1 both have the same number of squarefree divisors and powerful divisors.
2

%I #9 Feb 26 2023 03:26:39

%S 48,2511,5328,6723,7856,10287,15471,15632,16640,18063,20816,28592,

%T 33124,36368,38799,39600,40400,40816,54512,57121,60624,67472,75248,

%U 79375,83024,88047,93231,101168,119375,126927,134703,137456,146688,147824,148224,154448,160624

%N Numbers k such that k and k+1 both have the same number of squarefree divisors and powerful divisors.

%C Numbers k such that k and k+1 are both terms of A360902.

%H Amiram Eldar, <a href="/A360904/b360904.txt">Table of n, a(n) for n = 1..10000</a>

%e 48 is a term since A034444(48) = A005361(48) = 4 and A034444(49) = A005361(49) = 2.

%t q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Times @@ e == 2^Length[e]]; q[1] = True; seq[kmax_] := Module[{s = {}, k = 1, q1 = q[1], q2}, Do[q2 = q[k]; If[q1 && q2, AppendTo[s, k-1]]; q1 = q2, {k, 2, kmax}]; s]; seq[2*10^5]

%o (PARI) is(k) = {my(e = factor(k)[,2]); prod(i = 1, #e, e[i]) == 2^#e; }

%o lista(kmax) = {my(is1 = is(1), i2); for(k=2, kmax, is2 = is(k); if(is1 && is2, print1(k-1, ", ")); is1 = is2); }

%Y Subsequence of A360902.

%Y A360905 is a subsequence.

%Y Cf. A005361, A034444.

%K nonn

%O 1,1

%A _Amiram Eldar_, Feb 25 2023