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A137576 a(n) = A002326(n) * A006694(n) + 1. 18
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; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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 divises 2^d-2. Thus the sequence of such n is a subsequence of A050217.

LINKS

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, arXiv:0804.3682 [math.NT], 2008.

Vladimir Shevelev, Exact exponent in the remainder term of Gelfond's digit theorem in the binary case, Acta Arithmetica 136 (2009), 91-100.

FORMULA

It can be shown that if p is an odd prime then a((p^k-1)/2)=1+k*phi(p^k).

MATHEMATICA

a[n_] := (t = MultiplicativeOrder[2, 2n+1])*DivisorSum[2n+1, EulerPhi[#] / MultiplicativeOrder[2, #]&]-t+1; Table[a[n], {n, 0, 70}] (* Jean-Fran├žois Alcover, Dec 04 2015, adapted from PARI *)

PROG

(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

CROSSREFS

Cf. A002326, A006694, A138193, A138217, A138227, A141232, A195468.

Sequence in context: A195821 A208772 A071810 * A161329 A111745 A098957

Adjacent sequences:  A137573 A137574 A137575 * A137577 A137578 A137579

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Apr 26 2008, Apr 28 2008, May 03 2008, Jun 12 2008

EXTENSIONS

Edited and extended by Ray Chandler, May 08 2008

STATUS

approved

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Last modified November 24 00:27 EST 2017. Contains 295164 sequences.