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A319693 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.

%I A319693 #11 Oct 02 2018 17:41:23
%S A319693 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 A319693 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 A319693 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 A319693 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 A319693 Restricted growth sequence transform of A319692.
%C A319693 For all i, j: a(i) = a(j) => A305611(i) = A305611(j).
%H A319693 Antti Karttunen, <a href="/A319693/b319693.txt">Table of n, a(n) for n = 1..65537</a>
%e A319693 The proper divisors of  96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, while
%e A319693 the proper divisors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54.
%e A319693 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 A319693 (PARI)
%o A319693 up_to = 65537;
%o A319693 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 A319693 A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
%o A319693 A319692(n) = { my(m=1); fordiv(n, d, if(d<n, m *= prime(1+A001414(d)))); (m); };
%o A319693 v319693 = rgs_transform(vector(up_to,n,A319692(n)));
%o A319693 A319693(n) = v319693[n];
%Y A319693 Cf. A001414, A319692.
%Y A319693 Cf. also A319353.
%Y A319693 Differs from A305800, A296073 and A317943 for the first time at n=108, as here a(108) = 73.
%K A319693 nonn
%O A319693 1,2
%A A319693 _Antti Karttunen_, Oct 02 2018