| 1, 3, 5, 7, 13, 11, 13, 17, 17, 19, 31, 23, 41, 55, 29, 31, 41, 61, 37, 49, 41, 43, 85, 47, 85, 57, 53, 81, 73, 59, 61, 73, 73, 67, 111, 71, 73, 141, 151, 79, 217, 83, 89, 113, 89, 109, 131, 145, 97, 211, 101, 103, 169, 107, 109, 145, 113, 221, 133, 193, 221, 141, 301, 127
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Do there exist there composite numbers n for which a((n-1)/2)=n?
Theorem. If p and q are odd primes then the equality a((pq-1)/2)=pq is valid if and only if A002326((p-1)/2)=A002326((q-1)/2). Example: A002326(11) = A002326(44). Since 23 and 89 are primes then a((23*89-1)/2)=23*89.
A generalization: If p_1<p_2<...<p_m are distinct odd primes then a(((p_1*p_2*...*p_m)-1)/2)=p_1*p_2*...*p_m if and only if A002326((p_1-1)/2)= A002326((p_2-1)/2)=...=A002326((p_m-1)/2).
Moreover, if n is an odd square-free number and a((n-1)/2)=n then also all divisors d of n satisfy a((d-1)/2)=d and d divises 2^d-2. Thus the sequence of such n is a subsequence of A050217.
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LINKS
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Ray Chandler, Table of n, a(n) for n=0..10000
Vladimir Shevelev, Exact exponent of remainder term of Gelfond's digit theorem in binary case
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FORMULA
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It can be shown that if p is an odd prime then a((p^k-1)/2)=1+k*phi(p^k).
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CROSSREFS
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Cf. A002326, A006694, A138193, A138217 and A138227.
Sequence in context: A070334 A137700 A071810 this_sequence A111745 A098957 A018205
Adjacent sequences: A137573 A137574 A137575 this_sequence A137577 A137578 A137579
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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), Apr 26 2008, Apr 28 2008, May 03 2008, Jun 12 2008
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EXTENSIONS
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Edited and extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), May 08 2008
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