A173138 Composite numbers k such that 2^(k-4) == 1 (mod k).

4, 40369, 673663, 990409, 1697609, 2073127, 6462649, 7527199, 7559479, 14421169, 21484129, 37825753, 57233047, 130647919, 141735559, 179203369, 188967289, 218206489, 259195009, 264538057, 277628449, 330662479, 398321239, 501126487, 506958313, 612368311, 767983759

Offset: 1

Author

Michel Lagneau, Feb 10 2010

Keywords

• nonn

Comments

Besides the initial term, the sequence coincides with A033984 and consists of the odd terms > 7 of A015924.

Examples

			4 is a term: 2^(4 - 4) = 1 (mod 4).
		

References

• A. E. Bojarincev, Asymptotic expressions for the n-th composite number, Univ. Mat. Zap. 6:21-43 (1967). - In Russian.
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 2.

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..62

Crossrefs

Cf. A002808, A005381, A033984.

Programs

Maple

with(numtheory): for n from 1 to 100000000 do: a:= 2^(n-4)- 1; b:= a / n; c:= floor(b): if b = c and tau(n) <> 2 then print (n); else fi;od:

Mathematica

Select[Range[500000000],!PrimeQ[#]&&PowerMod[2,#-4,#]==1&] (* Harvey P. Dale, Nov 23 2011 *)

PARI

is(n)=!isprime(n) && n>1 && Mod(2,n)^(n-4)==1 \\ Charles R Greathouse IV, Nov 23 2011

Python

from sympy import isprime, prime, nextprime
def afind(k=4):
    while True:
        if pow(2, k-4, k) == 1 and not isprime(k): print(k, end=", ")
        k += 1
afind() # Michael S. Branicky, Mar 21 2022

Extensions

Simplified the definition, added cross-reference to A033984 R. J. Mathar, May 18 2010

More terms from Harvey P. Dale, Nov 23 2011

Typo in a(13) corrected by Georg Fischer, Mar 19 2022

a(24) and beyond from Michael S. Branicky, Mar 21 2022