%I #11 Oct 02 2018 17:41:23

%S 1,2,2,3,2,4,2,5,6,7,2,8,2,9,10,11,2,12,2,13,14,15,2,16,17,18,19,20,2,

%T 21,2,22,23,24,25,26,2,27,28,29,2,30,2,31,32,33,2,34,35,36,37,38,2,39,

%U 40,41,42,43,2,44,2,45,46,47,48,49,2,50,51,52,2,53,2,54,55,56,57,58,2,59,60,61,2,62,63,64,65,66,2,67,68,69,70,71,72,73,2,74,75,76,2,77,2,78,79,80,2,73,2,81,82,83,2,84,85

%N Filter sequence combining sopfr(d) from all proper divisors d of n, where sopfr(d) is A001414(d) = sum of primes dividing d with repetition.

%C Restricted growth sequence transform of A319692.

%C For all i, j: a(i) = a(j) => A305611(i) = A305611(j).

%H Antti Karttunen, <a href="/A319693/b319693.txt">Table of n, a(n) for n = 1..65537</a>

%e The proper divisors of 96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, while

%e the proper divisors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54.

%e It happens that sopfr(8) = sopfr(9), sopfr(16) = sopfr(18), sopfr(24) = sopfr(27), sopfr(32) = sopfr(36) and sopfr(48) = sopfr(54), and the rest of proper divisors (1, 2, 3, 4, 6, 12) are shared by both numbers, from which follows that by taking product of sopfr over proper divisors gives an identical result for both, thus a(96) = a(108). Here sopfr = A001414.

%o (PARI)

%o up_to = 65537;

%o 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 A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.

%o A319692(n) = { my(m=1); fordiv(n, d, if(d<n, m *= prime(1+A001414(d)))); (m); };

%o v319693 = rgs_transform(vector(up_to,n,A319692(n)));

%o A319693(n) = v319693[n];

%Y Cf. A001414, A319692.

%Y Cf. also A319353.

%Y Differs from A305800, A296073 and A317943 for the first time at n=108, as here a(108) = 73.

%K nonn

%O 1,2

%A _Antti Karttunen_, Oct 02 2018