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A296085 Filter sequence combining A296078(n) and A296092(n), the prime signatures of 1+phi(n) and 1+sigma(n).

%I A296085 #12 Feb 16 2025 08:33:52
%S A296085 1,2,1,3,1,1,2,4,5,1,1,1,5,2,6,7,1,8,5,9,5,1,2,9,10,1,1,5,1,9,5,11,12,
%T A296085 5,6,13,5,1,14,5,1,1,13,15,9,1,2,3,5,15,16,17,5,2,1,6,4,5,1,2,13,1,18,
%U A296085 19,14,15,5,16,20,14,1,21,13,5,3,5,1,6,4,15,15,1,5,21,16,5,12,1,5,14,1,22,5,5,2,15,13,13,1,5,1,15,18,9,9
%N A296085 Filter sequence combining A296078(n) and A296092(n), the prime signatures of 1+phi(n) and 1+sigma(n).
%C A296085 Restricted growth sequence transform of P(A296078(n), A296092(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.
%C A296085 For all i, j:
%C A296085   a(i) = a(j) => A296213(i) = A296213(j).
%H A296085 Antti Karttunen, <a href="/A296085/b296085.txt">Table of n, a(n) for n = 1..65537</a>
%H A296085 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%H A296085 Wikipedia, <a href="https://en.wikipedia.org/wiki/Pairing_function">Pairing Function</a>
%o A296085 (PARI)
%o A296085 up_to = 65537;
%o A296085 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 A296085 write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
%o A296085 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from _Charles R Greathouse IV_, Aug 17 2011
%o A296085 A296078(n) = A046523(1+eulerphi(n));
%o A296085 A296092(n) = A046523(1+sigma(n));
%o A296085 Anotsubmitted5(n) = (1/2)*(2 + ((A296078(n)+A296092(n))^2) - A296078(n) - 3*A296092(n));
%o A296085 write_to_bfile(1,rgs_transform(vector(up_to,n,Anotsubmitted5(n))),"b296085.txt");
%Y A296085 Cf. A000010, A000203, A046523, A296078, A296092, A296213.
%K A296085 nonn
%O A296085 1,2
%A A296085 _Antti Karttunen_, Dec 08 2017