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Numbers n such that the numbers of divisors of n,n+1,n+2 and n+3 are k,2k,4k,8k respectively for some k.
5

%I #8 Feb 25 2020 17:51:08

%S 613,757,997,1237,2917,6133,9853,15733,19477,22501,24421,25237,26293,

%T 26437,28117,31477,33301,35317,44101,45061,46357,51637,53077,56701,

%U 59221,61653,61717,64381,65917,66853,68583,70501,70887,72901,75133

%N Numbers n such that the numbers of divisors of n,n+1,n+2 and n+3 are k,2k,4k,8k respectively for some k.

%H Harvey P. Dale, <a href="/A100364/b100364.txt">Table of n, a(n) for n = 1..1000</a>

%e n=613,614,615,616 have 2,4,8,16 divisors respectively.

%t d0[x_] :=DivisorSigma[0, x];ta={{0}}; Do[g=n;s=d0[n];s1=d0[n+1];s2=d0[n+2];s3=d0[n+3] If[Equal[s1, 2*s]&&Equal[s2, 4*s]&&Equal[s3, 8*s], ta=Append[ta, n]; Print[{n, s, s1, s2, s3}]], {n, 1, 1000000}];{ta=Delete[ta, 1], g}

%t nd2Q[{a_,b_,c_,d_}]:={b,c,d}/a=={2,4,8}; Position[Partition[ DivisorSigma[ 0,Range[80000]],4,1],_?(nd2Q[#]&)]//Flatten (* _Harvey P. Dale_, Feb 25 2020 *)

%Y Cf. A000005, A063446, A100363.

%K nonn

%O 1,1

%A _Labos Elemer_, Nov 19 2004