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Search: id:A141216
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| 30, 320, 224, 240, 72, 360, 728, 0, 672, 216, 1320, 0, 0, 16, 5060, 60, 126, 10560, 216, 0, 3360, 2574, 150
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OFFSET
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1,1
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COMMENT
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The zero terms are of a special interest. Indeed, since for any odd prime p, A137576((p-1)/2)=p, then it is natural to call "overpseudoprimes" those Poulet pseudoprimes A001567(n) for which a(n)=0.
Theorem. A square-free composite number m = p_1*p_2*...*p_k is an overpseudoprime if and only if A002326((p_1-1)/2)=A002326((p_2-1)/2)=...=A002326((p_k-1)/2). Moreover, every overpseudoprime is in A001262.
Note that in A001262 there exist terms which are not square-free. The first is A001262(52)=1194649 =1093^2.
It can be shown that if an overpseudoprime is not a multiple of the square of a Wieferich prime (see A001220) then it is squarefree. Also all squares of Wieferich primes are overpseudoprimes.
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REFERENCES
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V. Shevelev, Overpseudoprimes, Mersenne Numbers and Wieferich Primes, arxiv.org/abs/0806.3412
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CROSSREFS
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Cf. A137576, A001567, A001262, A002326, A006694.
Sequence in context: A042750 A074994 A134287 this_sequence A159543 A006859 A107967
Adjacent sequences: A141213 A141214 A141215 this_sequence A141217 A141218 A141219
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KEYWORD
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nonn
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AUTHOR
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Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jun 14 2008, Jul 13 2008
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