A121707 -id:A121707 - cp's OEIS frontend

A318055 Numbers k such that gcd(k, 2^k - 2) = 1 and gcd(k, 3^k - 3) > 1.

247, 403, 559, 715, 871, 1027, 1339, 1495, 1651, 1807, 1963, 2009, 2035, 2119, 2587, 2743, 2899, 2993, 3055, 3211, 3523, 3649, 3679, 3835, 3977, 3991, 4147, 4303, 4331, 4453, 4615, 4633, 4699, 4771, 4927, 5239, 5395, 5617, 5707, 5863, 5995, 6019, 6031, 6161, 6331, 6487, 6799, 6929, 6955, 7081, 7111

Offset: 1

Thomas Ordowski, Aug 14 2018

nonn

Odd numbers k such that gcd(k,2^(k-1)-1) = 1 and gcd(k,3^(k-1)-1) > 1.
It seems that a(n) == 91 (mod 156) for infinitely many n.
Fermat pseudoprimes to base 3 (A005935) in this sequence are 16531, 49051, 72041, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A267999 and probably of A121707.
Cf. A139613(2n+1): it gives many terms of the sequence.
Cf. A005935.

Filtered([1..10000],k->Gcd(k,2^k-2) = 1 and Gcd(k,3^k-3) > 1);  # Muniru A Asiru, Oct 07 2018

select(k->gcd(k,2^k-2) = 1 and gcd(k,3^k-3) > 1,[$1..10000]); # Muniru A Asiru, Oct 07 2018

Select[Range[8000], GCD[#, 2^# - 2] == 1 && GCD[#, 3^# - 3] > 1 &] (* Amiram Eldar, Mar 31 2024 *)

isok(k) = (gcd(k,2^k-2) == 1) && (gcd(k,3^k-3) != 1); \\ Michel Marcus, Aug 14 2018

More terms from Michel Marcus, Aug 14 2018

A326584 a(n) = gcd(n*N(n-1), D(n-1)), with N(n)/D(n) = B(n) the n-th Bernoulli number.

Original entry on oeis.org

1, 2, 3, 1, 5, 1, 7, 1, 3, 1, 11, 1, 13, 1, 3, 1, 17, 1, 19, 1, 3, 1, 23, 1, 5, 1, 3, 1, 29, 1, 31, 1, 3, 1, 1, 1, 37, 1, 3, 1, 41, 1, 43, 1, 15, 1, 47, 1, 7, 1, 3, 1, 53, 1, 1, 1, 3, 1, 59, 1, 61, 1, 3, 1, 5, 1, 67, 1, 3, 1, 71, 1, 73, 1, 3, 1, 1, 1, 79, 1

Offset: 1

Peter Luschny, Jul 19 2019

nonn

Conjectures:
(1) If n > 1 then a(n) = n <=> n is prime or Carmichael (A002997).
(2) If n is odd then a(n) = 1 <=> n = 1 or is a term of A121707.
(3) The fixed points of n^2/a(n) are exactly the numbers satisfying Korselt's criterion (compare A326578 and A324050).

			a(559) =   1 and 559 is in A121707.
a(561) = 561 and 561 is Carmichael.
a(563) = 563 and 563 is prime.

Peter Luschny, Table of n, a(n) for n = 1..10000

Cf. A000040, A002997, A121707, A027641/A027642 (Bernoulli), A324050 (Korselt).

Cf. A326578, A326478, A326579, A326577.

db := n -> denom(bernoulli(n)): nb := n -> numer(bernoulli(n)):
a := n -> igcd(n*nb(n-1), db(n-1)): seq(a(n), n=1..80);

a[n_] := With[{b = BernoulliB[n-1]}, GCD[n Numerator[b], Denominator[b]]];
Array[a, 80] (* Jean-François Alcover, Jul 21 2019 *)

a(n) = my(b=bernfrac(n-1)); gcd(n*numerator(b), denominator(b)); \\ Michel Marcus, Jul 19 2019

a(n) divides n, n/a(n) = A326478(n).

A316908 a(n) is the smallest k with n prime factors such that 2^(k-1) == 1 (mod k) and p-1 does not divide k-1 for every prime p dividing k.

Original entry on oeis.org

7957, 617093, 134564501, 384266404601, 8748670222601, 6105991025919737, 901196605940857381

Offset: 2

Thomas Ordowski, Jul 16 2018

Conjecture: a(n) > A006931(n) for every n > 2.
a(6)-a(8) derived from Feitsma's tables of pseudoprimes. a(9) > 2^64. - Giovanni Resta, Jul 19 2018
From Daniel Suteu, Jun 08 2020: (Start)
a(9)  <= 521957994426556057126261,
a(10) <= 1315856103949347820015303981,
a(11) <= 6357507186189933506573017225316941,
a(12) <= 77822245466150976053960303855104674781. (End)

Jan Feitsma and William Galway, Tables of pseudoprimes and related data

Cf. A001567, A121707, A316907.

More terms from Michel Marcus, Jul 16 2018

a(6)-a(8) from Giovanni Resta, Jul 19 2018

A319484 a(n) is the smallest k > 1 such that n^k == n (mod k) and gcd(k, b^k-b) = 1 for some b <> n.

Original entry on oeis.org

35, 35, 7957, 16531, 1247, 4495, 35, 817, 2501, 697, 55, 55, 143, 221, 35, 35, 1247, 493, 221, 95, 35, 35, 77, 253, 115, 403, 247, 247, 203, 35, 155, 155, 697, 187, 35, 35, 35, 589, 221, 95, 533, 35, 287, 77, 55, 55, 115, 221, 329, 35, 35, 221, 221, 689, 55, 35, 35

Offset: 0

Thomas Ordowski, Oct 26 2018

nonn

a(n) is the smallest k > 1 such that n^k == n (mod k) and p-1 does not divide k-1 for every prime p dividing k, see A121707.
We have A000790(n) < a(n) <= A316940(n) for n > 0.
It seems that the sequence is unbounded like A316940.
The term a(5) = 4495 = 5*29*31 is not semiprime.

			a(6) = 35 since 6^35 == 6 (mod 35) and 35 = 5*7 is the smallest "anti-Carmichael number": 5-1 does not divide 7-1. We have gcd(35,2^35-2) = 1.

Cf. A000790, A121707, A316940.

isac(n) = {my(f = factor(n)[,1]); for (i=1, #f, if (((n-1) % (f[i]-1)) == 0, return (0));); return (1);}
isok(n,k) = {if (Mod(n, k)^k != Mod(n, k), return (0)); return (isac(k));}
a(n) = {my(k=2); while (!isok(n,k), k++); return (k);} \\ Michel Marcus, Oct 27 2018

More terms from Michel Marcus, Oct 26 2018

A345675 Numbers m such that D_{m-1} is the smallest base b > 1 for which b^{m-1} == 1 (mod m), where D_k is the denominator (A027642) of Bernoulli number B_k.

Original entry on oeis.org

35, 14315, 22399, 35711, 455891, 881809, 1198159, 1917071, 2287987, 3310037, 4464941, 11029439, 12190061, 13325753, 17832803, 33012941, 33296147, 37814849, 44986423, 74437181, 76911149, 82873661, 91909571, 98859851, 108266171, 128008159, 128981243, 132391409

Offset: 1

Thomas Ordowski, Sep 04 2021

nonn

These are numbers m such that A027642(m-1) = A105222(m).

The corresponding bases of these pseudoprimes are 6, 6, 42, 66, 66, 46410, 3318, 66, 42, 30, 330, 6, 330, 61410, 6, 330, 1074, 510, 3318, 330, 7890, 330, 66, 12606, 66, 42, 6, 510, ...

Cf. A027642, A105222, A121707, A316940.

Den[n_] := Times @@ (1 + Select[Divisors[n], PrimeQ[# + 1] &]); q[k_] := Module[{m = 2, d = Den[k - 1]}, If[PowerMod[d, k - 1, k] != 1, False, While[m < d && PowerMod[m, k - 1, k] != 1, m++]; m == d]]; Select[Range[3, 10^6, 2], q] (* Amiram Eldar, Sep 04 2021 *)

f(n) = my(m=2); while(Mod(m, n)^(n-1)!=1, m++); m;
isok(m) = f(m) == denominator(bernfrac(m-1)); \\ Michel Marcus, Sep 04 2021

More terms from Amiram Eldar, Sep 04 2021

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A318055 Numbers k such that gcd(k, 2^k - 2) = 1 and gcd(k, 3^k - 3) > 1.

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A326584 a(n) = gcd(n*N(n-1), D(n-1)), with N(n)/D(n) = B(n) the n-th Bernoulli number.

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A316908 a(n) is the smallest k with n prime factors such that 2^(k-1) == 1 (mod k) and p-1 does not divide k-1 for every prime p dividing k.

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A319484 a(n) is the smallest k > 1 such that n^k == n (mod k) and gcd(k, b^k-b) = 1 for some b <> n.

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A345675 Numbers m such that D_{m-1} is the smallest base b > 1 for which b^{m-1} == 1 (mod m), where D_k is the denominator (A027642) of Bernoulli number B_k.

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