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Define the total signature symmetry of a number n to be the number of values r takes such that n-r and n+r have the same prime signature. Sequence contains the total signature symmetry of n.
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%I #13 Sep 01 2018 09:21:05

%S 0,0,0,1,1,1,1,3,2,3,2,4,2,3,4,5,4,7,2,6,4,5,4,10,5,5,7,7,4,10,4,10,7,

%T 7,8,16,6,7,10,11,4,14,7,10,13,9,8,18,5,13,11,12,8,20,11,17,11,13,8,

%U 27,5,10,17,16,11,19,10,14,11,15,15,33,10,15,20,15,12,23,9,22,18,13,12,31,14

%N Define the total signature symmetry of a number n to be the number of values r takes such that n-r and n+r have the same prime signature. Sequence contains the total signature symmetry of n.

%C Number of partitions of 2n in two parts with identical prime signatures. Conjecture: (1) No term is zero for n > 3. (2) Every number k appears a finitely many times in the sequence. I.e., for every k there exists a number f(k) such that for all n > f(k), a(n) > k. Subsidiary sequence: the frequency of n.

%H Charlie Neder, <a href="/A093489/b093489.txt">Table of n, a(n) for n = 1..1000</a>

%e a(12) = 4. The four values r can take are 1,2,5 and 7, giving the four pairs of numbers with identical prime signatures; (11,13),(10,14),(7,17) and (5,19).

%Y Cf. A093488, A093491, A093492.

%K nonn,easy

%O 1,8

%A _Amarnath Murthy_, Apr 16 2004

%E More terms from _Joshua Zucker_, Jul 24 2006