A173572 Odd integers n such that 2^n == 4 (mod n).

1, 20737, 93527, 228727, 373457, 540857, 2231327, 11232137, 15088847, 15235703, 24601943, 43092527, 49891487, 66171767, 71429177, 137134727, 207426737, 209402327, 269165561, 302357057, 383696711, 513013327

Offset: 1

Author

Michel Lagneau, Feb 22 2010

Keywords

• nonn

Comments

The odd terms of A015921.

Also, nonprime integers n such that 2^(n-2) == 1 (mod n).

For all m, 2^A050259(m)-1 belongs to this sequence.

If n > 1 is a term and p is a primitive prime factor of 2^(n-2)-1, then n*p is also a term. Hence, the sequence is infinite. (Rotkiewicz 1984)

References

• A. E. Bojarincev, Asymptotic expressions for the n-th composite number, Univ. Mat. Zap. 6:21-43 (1967). (in Russian)
• R. K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, Third Edition, 2004
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 2.

Links

• Max Alekseyev, Table of n, a(n) for n = 1..722 (all terms below 10^14)
• C. K. Caldwell, Composite Numbers
• A. Rotkiewicz, On the congruence 2^(n-2) == 1 (mod n). Math. Comp. 43 (1984), 271-272.

Crossrefs

Cf. A015921, A050259, A173138, A002808.

Programs

Maple

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

Mathematica

m = 4; Join[Select[Range[1, m, 2], Divisible[2^# - m, #] &], Select[Range[m + 1, 10^6, 2], PowerMod[2, #, #] == m &]] (* Robert Price, Oct 12 2018 *)

PARI

is(n) = n%2==1 && Mod(2,n)^n==Mod(4,n) \\ Jinyuan Wang, Feb 22 2019

Extensions

Edited and term 1 prepended by Max Alekseyev, Aug 09 2012