A291855 -id:A291855 - cp's OEIS frontend

A292559 Composite numbers m such that lpf(2^m - 1) == 1 (mod m).

169, 221, 323, 611, 779, 793, 923, 1121, 1159, 1271, 1273, 1343, 1349, 1513, 1717, 1829, 1919, 2033, 2077, 2201, 2413, 2533, 2603, 2759, 2951, 3097, 3131, 3173, 3193, 3281, 3379, 3599, 3721, 3743, 3791, 3937, 3953, 4043, 4223, 4309, 4331, 4369, 4607, 4619, 4811, 4867, 4883, 4981, 5111, 5177, 5263, 5429, 5567, 5699

Offset: 1

Michel Marcus and Thomas Ordowski, Sep 19 2017

All terms are coprime to 2, 3, 5, 7, 11. - Robert Israel, Sep 20 2017
If p = lpf(2^m - 1) and A002326((p-1)/2) = m composite, then m is in this sequence. - Thomas Ordowski, Sep 20 2017
Conjecture: there are no numbers k in this sequence such that, for each prime factor q of 2^k - 1, q == 1 (mod k). - Thomas Ordowski, Sep 20 2017
Note: if all prime factors q of 2^k - 1 are q == 1 (mod k), then 2^k - 1 == 1 (mod k), thus 2^k == 2 (mod k), so k is a pseudoprime. The pseudoprime k = a(42) = 4369 = 17*257 is not a counterexample to this conjecture. A pseudoprime k = P*Q such that both 2^P - 1 and 2^Q - 1 are primes would be a counterexample, but the known Mersenne primes do not give such k. - Thomas Ordowski, Oct 02 2017
If lpf(2^n - 1) == 1 (mod n), then gpf(2^n - 1) == 1 (mod n). Cf. A291855. - Thomas Ordowski, Oct 20 2017
Composites m such that lpf(2^m - 1)*gpf(2^m - 1) is a Fermat pseudoprime to base 2, i.e., is in A214305. - Thomas Ordowski, Oct 29 2017

Subsequence of A236769.

Cf. A002326, A049479, A292834, A298076.

searchMax = 1000; Complement[Select[Range[searchMax], Mod[FactorInteger[2^# - 1][[1, 1]], #] == 1 &], Prime[Range[PrimePi[searchMax]]]] (* Alonso del Arte, Sep 19 2017 *)

lista(nn) = forcomposite(n=1, nn, sp = factor(2^n-1)[1,1]; if ((sp % n) == 1, print1(n, ", "))); \\ Michel Marcus, Sep 19 2017

A049479(m) == 1 (mod m).

a(10)-a(54) from Charles R Greathouse IV, Sep 19 2017

A292199 Numbers m such that there is a prime divisor p > m of 2^m - 1 not of the form k*m + 1.

Original entry on oeis.org

20, 25, 27, 28, 33, 35, 44, 45, 48, 49, 52, 54, 55, 56, 60, 63, 66, 68, 69, 70, 72, 76, 77, 80, 81, 84, 87, 88, 90, 91, 92, 95, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 116, 117, 119, 120, 123, 124, 125, 126, 129, 132, 135, 136, 138, 140, 141, 143, 144, 145, 147, 148, 150

Offset: 1

Thomas Ordowski, Sep 11 2017

nonn

Contains no primes q and no doubled primes 2q.

Are there infinitely many such numbers?

Max Alekseyev, Table of n, a(n) for n = 1..829

Cf. A000040, A000225, A060443, A292188.

Contains A291855.

Terms a(19) onward from Max Alekseyev, Sep 13 2017

cp's OEIS Frontend

A292559 Composite numbers m such that lpf(2^m - 1) == 1 (mod m).

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