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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
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OFFSET
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0,2
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COMMENTS
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Composite numbers n for which a((n-1)/2)=n are called overpseudoprimes to base 2 (A141232).
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 squarefree number and a((n-1)/2)=n then also all divisors d of n satisfy a((d-1)/2)=d and d divides 2^d-2. Thus the sequence of such n is a subsequence of A050217.
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LINKS
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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).
a(n) = ord(2,2*n+1) * ((Sum_{d|(2n+1)} phi(d)/ord(2,d)) - 1) + 1, where phi = A000010 and ord(2,d) is the multiplicative order of 2 modulo d. - Jianing Song, Nov 13 2021
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MATHEMATICA
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a[n_] := (t = MultiplicativeOrder[2, 2n+1])*DivisorSum[2n+1, EulerPhi[#] / MultiplicativeOrder[2, #]&]-t+1; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Dec 04 2015, adapted from PARI *)
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PROG
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(PARI) a(n)=my(t); sumdiv(2*n+1, d, eulerphi(d)/(t=znorder(Mod(2, d))))*t-t+1 \\ Charles R Greathouse IV, Feb 20 2013
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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