A291194 Numbers k having at least one prime factor p such that p^2 divides 2^(k-1) - 1.

1093, 3511, 398945, 796797, 1194649, 1592501, 1990353, 2388205, 2786057, 3183909, 3581761, 3979613, 4377465, 4775317, 5173169, 5571021, 5968873, 6165316, 6366725, 6764577, 7162429, 7560281, 7958133, 8355985, 8753837, 9151689, 9549541, 9947393, 10345245

Offset

1,1

Comments

Another version of A001220.

Sequence is infinite since if k is a term then also k^m is a term, for every m >= 2.

What is the smallest number in this sequence which is not of the form 13*n + 1?

Complete factorizations of the first 15 terms:

a(1) = 1093

a(2) = 3511

a(3) = 5 * 73 * 1093

a(4) = 3^6 * 1093

a(5) = 1093^2

a(6) = 31 * 47 * 1093

a(7) = 3 * 607 * 1093

a(8) = 5 * 19 * 23 * 1093

a(9) = 1093 * 2549

a(10) = 3 * 971 * 1093

a(11) = 29 * 113 * 1093

a(12) = 11 * 331 * 1093

a(13) = 3^2 * 5 * 89 * 1093

a(14) = 17 * 257 * 1093

a(15) = 1093 * 4733

These are the numbers k for which gcd(k^2, 2^(k-1)-1) is not squarefree. However, numbers k such that gcd(k^2, 2^(k-1)-1) > k are a proper subset of them. Are there infinitely many such numbers? See A331021. - Amiram Eldar and Thomas Ordowski, Jan 06 2020

Links

Giovanni Resta, Table of n, a(n) for n = 1..10000

Prog

(Magma) lst:=[]; for n in [2..10345245] do f:=Factorization(n); if not IsNull([x: x in [1..#f] | Modexp(2, n-1, f[x][1]^2) eq 1]) then Append(~lst, n); end if; end for; lst;

Crossrefs

Cf. A190991, A270833. A001220 gives the primes.

Sequence in context: A001220 A265630 A355545 * A331021 A270833 A273471

Adjacent sequences: A291191 A291192 A291193 * A291195 A291196 A291197

Keyword

nonn

Author

Arkadiusz Wesolowski, Aug 20 2017

Status

approved