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

1,1

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.

Links

Table of n, a(n) for n=1..46.

Formula

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

Mathematica

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

Crossrefs

Cf. A001220, A001358.

Sequence in context: A102275 A299448 A300086 * A322056 A059880 A144096

Adjacent sequences: A115945 A115946 A115947 * A115949 A115950 A115951

Keyword

easy,nonn

Author

Jonathan Vos Post, Mar 14 2006

Extensions

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

Status

approved