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A319706 Filter sequence which for primes p records the prime signature of 2p+1, and for all other numbers assigns a unique number.

%I A319706 #9 Sep 26 2018 16:24:35
%S A319706 1,2,2,3,2,4,5,6,7,8,2,9,10,11,12,13,5,14,5,15,16,17,2,18,19,20,21,22,
%T A319706 2,23,24,25,26,27,28,29,24,30,31,32,2,33,5,34,35,36,5,37,38,39,40,41,
%U A319706 2,42,43,44,45,46,5,47,5,48,49,50,51,52,53,54,55,56,5,57,24,58,59,60,61,62,5,63,64,65,2,66,67,68,69,70,2,71,72,73,74,75,76,77,78,79,80,81,5
%N A319706 Filter sequence which for primes p records the prime signature of 2p+1, and for all other numbers assigns a unique number.
%C A319706 Restricted growth sequence transform of function f defined as f(n) = A046523(2n+1) when n is a prime, otherwise -n.
%C A319706 For all i, j:
%C A319706   A305810(i) = A305810(j) => a(i) = a(j),
%C A319706 and
%C A319706   a(i) = a(j) => A305800(i) = A305800(j),
%C A319706   a(i) = a(j) => A305978(i) = A305978(j),
%C A319706   a(i) = a(j) => A305985(i) = A305985(j).
%H A319706 Antti Karttunen, <a href="/A319706/b319706.txt">Table of n, a(n) for n = 1..100000</a>
%o A319706 (PARI)
%o A319706 up_to = 100000;
%o A319706 rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
%o A319706 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
%o A319706 A319706aux(n) = if(isprime(n),A046523(n+n+1),-n);
%o A319706 v319706 = rgs_transform(vector(up_to,n,A319706aux(n)));
%o A319706 A319706(n) = v319706[n];
%Y A319706 Cf- A046523, A305800, A305810, A305900, A305901, A305978, A305985.
%Y A319706 Cf. A005384 (positions of 2's), A234095 (positions of 5's).
%K A319706 nonn
%O A319706 1,2
%A A319706 _Antti Karttunen_, Sep 26 2018