A277185 Odd numbers k such that d(k) divides 2^k-1.

1, 729, 85766121, 1340095640625, 151939915084881, 413976684737889, 2070185663499849, 4034942722397601, 12696463968316569, 51015688922507841, 55593461341979649, 76117748092591401, 220052401647189489, 407398015096219161, 542158788145462929, 924491486192068809

Offset

1,2

Comments

Subsequence of odd terms of A277285.

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

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..38

Example

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

Mathematica

Select[Range[1000], Mod[ 2^# - 1, DivisorSigma[0, # ]] == 0 && OddQ[#] &] (* G. C. Greubel, Oct 18 2016 *)
Select[Range[1, 1.2*10^6, 2]^2, # == 1 || PowerMod[2, #, DivisorSigma[0, #]] == 1 &] (* Amiram Eldar, May 12 2024 *)

Prog

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

Crossrefs

Cf. A000005, A000225, A277285.

Subsequence of A016754.

Sequence in context: A017430 A017562 A013852 * A343693 A357960 A352034

Adjacent sequences: A277182 A277183 A277184 * A277186 A277187 A277188

Keyword

nonn

Author

Altug Alkan, Oct 03 2016

Extensions

a(4)-a(9) from Giovanni Resta, Oct 03 2016

a(10)-a(16) from Amiram Eldar, May 12 2024

Status

approved