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Smallest prime in representation 2*A241757(n) by sum of two primes, the adding of which in binary requires only one carry.
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%I #17 May 07 2014 00:30:23

%S 2,5,13,5,17,17,5,17,5,5,13,17,5,13,5,17,37,17,5,13,17,17,29,37,41,5,

%T 5,17,13,5,13,17,5,37,41,17,5,73,5,89,13,97,5,13,17,37,41,29,137,5,5,

%U 41,5,41,13,193,5,5,17,193,17,17,37,41,37,97,53,73,53,5

%N Smallest prime in representation 2*A241757(n) by sum of two primes, the adding of which in binary requires only one carry.

%H Peter J. C. Moses, <a href="/A241758/b241758.txt">Table of n, a(n) for n = 1..10000</a>

%H Aviezri Fraenkel and Alex Kontorovich, <a href="http://www.emis.de/journals/INTEGERS/papers/a14int2005/a14int2005.Abstract.html">The Sierpiński Sieve of Nim-varieties and Binomial Coefficients</a>, INTEGERS 7 (2)(2007), #A14.

%H E. E. Kummer, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0044&amp;IDDOC=266967">Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen</a>, J. Reine Angew. Math. 44 (1852), 93-146.

%F 2||Binomial(2*A241757(n), a(n)). Indeed, from the Kummer theorem (see reference) 2^t||Binomial(n,x) if and only if in adding x and n-x in binary we have exactly t carries. A proof of the Kummer theorem in arbitrary base one can find in [Fraenkel & Kontorovich].

%e a(2)=5, since A241757(2)=22=5+17, and in binary in sum of 101+10001 involves only one carry.

%Y Cf. A241757, A241405.

%K nonn,base

%O 1,1

%A _Vladimir Shevelev_, Apr 28 2014

%E More terms from _Peter J. C. Moses_, Apr 29 2014