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Odd numbers k such that d(k) divides 2^k-1.
2

%I #33 May 12 2024 00:46:24

%S 1,729,85766121,1340095640625,151939915084881,413976684737889,

%T 2070185663499849,4034942722397601,12696463968316569,

%U 51015688922507841,55593461341979649,76117748092591401,220052401647189489,407398015096219161,542158788145462929,924491486192068809

%N Odd numbers k such that d(k) divides 2^k-1.

%C Subsequence of odd terms of A277285.

%C Sequence is infinite. In particular, k = (21p)^6 for primes p = 5 or p > 7 are such: d(k) = 7^3 divides 2^147 - 1, which in turn divides 2^k - 1. - _Max Alekseyev_, 30 Sep 2016

%C Note that there are also other forms. For example, (A002110(6)/2)^6 is a term.

%H Amiram Eldar, <a href="/A277185/b277185.txt">Table of n, a(n) for n = 1..38</a>

%e Odd number 729 is a term because 2^729-1 is divisible by d(729) = 7.

%t Select[Range[1000], Mod[ 2^# - 1, DivisorSigma[0, # ]] == 0 && OddQ[#] &] (* _G. C. Greubel_, Oct 18 2016 *)

%t Select[Range[1, 1.2*10^6, 2]^2, # == 1 || PowerMod[2, #, DivisorSigma[0, #]] == 1 &] (* _Amiram Eldar_, May 12 2024 *)

%o (PARI) is(n) = (2^n-1) % numdiv(n) == 0 && n % 2 == 1;

%o (PARI) lista(kmax) = {my(d); forstep(k = 1, kmax, 2, d = vecprod(apply(x -> 2*x+1, factor(k)[,2])); if(Mod(2, d)^(k^2) == 1, print1(k^2, ", ")));} \\ _Amiram Eldar_, May 12 2024

%Y Cf. A000005, A000225, A277285.

%Y Subsequence of A016754.

%K nonn

%O 1,2

%A _Altug Alkan_, Oct 03 2016

%E a(4)-a(9) from _Giovanni Resta_, Oct 03 2016

%E a(10)-a(16) from _Amiram Eldar_, May 12 2024