A115948 a(n) = (2^(semiprime(n)-1)) modulo (semiprime(n)^2).

8, 32, 13, 12, 156, 184, 319, 464, 341, 496, 301, 308, 9, 952, 472, 508, 1191, 922, 2359, 688, 1800, 2668, 2291, 3109, 2888, 4860, 412, 4691, 604, 2875, 4523, 2236, 3856, 5659, 2016, 8662, 3259, 8852, 13239, 6953, 1344, 6277, 7357, 2857, 11660, 18193

Offset: 1

Author

Jonathan Vos Post, Mar 14 2006

Keywords

• easy
• nonn

Comments

Wieferich function of semiprimes.

This appears in the search for the semiprime analogy to A001220 Wieferich primes p: p^2 divides 2^(p-1) - 1. That is, the Wieferich function W(p) of primes p is W(p) = 2^(p-1) modulo p^2 and a (rare!) Wieferich prime (A001220) is one such that W(p) = 1. The current sequence is W(semiprime(n)). Any semiprime s for which W(s) = 1 would be a "Wieferich semiprime." This is also related to Fermat's "little theorem" that for any odd prime p we have 2^(p-1) == 1 modulo p.

Such a "Wieferich semiprime" would be a special case of a "Wieferich pseudoprime", i.e. it would be a composite integer that is one more than a term in A240719 and has two prime factors. - Felix Fröhlich, Jul 16 2014

References

• R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 28.
• R. K. Guy, Unsolved Problems in Number Theory, A3.
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 91.

Crossrefs

Cf. A001220, A001358.

Programs

Mathematica

PowerMod[2, # - 1, #^2] & /@ Select[ Range@141, Plus @@ Last /@ FactorInteger@# == 2 &] (* Robert G. Wilson v *)

Formula

a(n) = (2^(A001358(n)-1)) modulo (A001358(n)^2).

Extensions

More terms from Robert G. Wilson v, Mar 14 2006