A316940 -id:A316940 - cp's OEIS frontend

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.

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

cp's OEIS Frontend

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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