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%I #13 Jul 20 2014 00:19:28
%S 8,32,13,12,156,184,319,464,341,496,301,308,9,952,472,508,1191,922,
%T 2359,688,1800,2668,2291,3109,2888,4860,412,4691,604,2875,4523,2236,
%U 3856,5659,2016,8662,3259,8852,13239,6953,1344,6277,7357,2857,11660,18193
%N a(n) = (2^(semiprime(n)-1)) modulo (semiprime(n)^2).
%C Wieferich function of semiprimes.
%C 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.
%C 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
%D R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 28.
%D R. K. Guy, Unsolved Problems in Number Theory, A3.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 91.
%F a(n) = (2^(A001358(n)-1)) modulo (A001358(n)^2).
%t PowerMod[2, # - 1, #^2] & /@ Select[ Range@141, Plus @@ Last /@ FactorInteger@# == 2 &] (* _Robert G. Wilson v_ *)
%Y Cf. A001220, A001358.
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Mar 14 2006
%E More terms from _Robert G. Wilson v_, Mar 14 2006
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