A331021 -id:A331021 - cp's OEIS frontend

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

Arkadiusz Wesolowski, Aug 20 2017

nonn

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

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

Cf. A190991, A270833. A001220 gives the primes.

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;

cp's OEIS Frontend

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

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