A180942 Odd composite numbers m for which A000111(m) == (-1)^( (m-1)/2 ) (mod m).

91, 561, 781, 1105, 1661, 1729, 2465, 2737, 2821, 6601, 8911, 10585, 15841, 29341, 30433, 41041, 46657, 52633, 62745, 63973, 75361, 90241, 101101, 115921, 126217, 136371, 136741, 137149, 162401, 172081, 176565, 188461, 251251, 252601, 278545, 294409, 314821, 334153

Offset: 1

Author

Vladimir Shevelev, Sep 27 2010

Keywords

• nonn

Comments

For any odd prime p, A000111(p) == (-1)^((p-1)/2) mod p, see A180418, so these cases are not considered further and left out of the sequence by definition.

Might be called "Zig-zag pseudoprimes."

It seems that every Carmichael number (A002997) <= 512461 is in the sequence. - D. S. McNeil, Sep 01 2010

Links

• Vladimir Shevelev, The number of permutations with prescribed up-down structure as a function of two variables, INTEGERS 12 (2012), article #A1. - From _N. J. A. Sloane_, Feb 07 2013

Crossrefs

Cf. A000111.

Programs

Mathematica

fQ[n_] := ! PrimeQ@n && Mod[ (-1)^((n - 1)/2)*2^(n + 1)*(2^(n + 1) - 1)*BernoulliB[n + 1]/(n + 1), n] == Mod[(-1)^((n - 1)/2), n]; k = 3; lst = {}; While[k < 50000, If[ fQ@k, AppendTo[lst, k]; Print@k]; k += 2]; lst (* Robert G. Wilson v, Sep 29 2010 *)

Python

from itertools import count, islice, accumulate
from sympy import isprime
def A180942_gen(): # generator of terms
    blist = (0,1)
    for n in count(2):
        blist = tuple(accumulate(reversed(blist),initial=0))
        if n & 1 and (blist[-1] + (1 if (n-1)//2 & 1 else -1)) % n == 0 and not isprime(n):
            yield n
A180942_list = list(islice(A180942_gen(),5)) # Chai Wah Wu, Jun 09-11 2022

Extensions

Extended to a(13) by D. S. McNeil, Sep 01 2010

Comments rephrased by R. J. Mathar, Sep 29 2010

a(14)-a(17) from Robert G. Wilson v, Sep 29 2010

a(18)-a(38) from Amiram Eldar, Dec 28 2019