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A319357 Filter sequence combining A003415(d) from all proper divisors d of n, where A003415(d) = arithmetic derivative of d.

%I A319357 #14 Oct 03 2018 21:35:37
%S A319357 1,2,2,3,2,4,2,5,3,4,2,6,2,4,4,7,2,8,2,9,4,4,2,10,3,4,11,12,2,13,2,14,
%T A319357 4,4,4,15,2,4,4,16,2,17,2,18,19,4,2,20,3,21,4,22,2,23,4,24,4,4,2,25,2,
%U A319357 4,26,27,4,28,2,29,4,30,2,31,2,4,32,33,4,34,2,35,36,4,2,37,4,4,4,38,2,39,4,40,4,4,4,41,2,42,43,44,2,45,2,46,47
%N A319357 Filter sequence combining A003415(d) from all proper divisors d of n, where A003415(d) = arithmetic derivative of d.
%C A319357 Restricted growth sequence transform of A319356.
%C A319357 The only duplicates in range 1..65537 with a(n) > 4 are the following six pairs: a(1445) = a(2783), a(4205) = a(11849), a(5819) = a(8381), a(6727) = a(15523), a(8405) = a(31211) and a(28577) = a(44573). All these have prime signature p^2 * q^1. If all the other duplicates respect the prime signature as well, then also the last implication given below is valid.
%C A319357 For all i, j:
%C A319357   a(i) = a(j) => A000005(i) = A000005(j),
%C A319357   a(i) = a(j) => A319683(i) = A319683(j),
%C A319357   a(i) = a(j) => A319686(i) = A319686(j),
%C A319357   a(i) = a(j) => A101296(i) = A101296(j). [Conjectural, see notes above]
%H A319357 Antti Karttunen, <a href="/A319357/b319357.txt">Table of n, a(n) for n = 1..65537</a>
%e A319357 Proper divisors of 1445 are [1, 5, 17, 85, 289], while the proper divisors of 2783 are [1, 11, 23, 121, 253]. 1 contributes 0 and primes contribute 1, so only the last two matter in each set. We have A003415(85) = 22 = A003415(121) and A003415(289) = 34 = A003415(253), thus the value of arithmetic derivative coincides for all proper divisors, thus a(1445) = a(2783).
%o A319357 (PARI)
%o A319357 up_to = 65537;
%o A319357 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 A319357 A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
%o A319357 A319356(n) = { my(m=1); fordiv(n, d, if(d<n, m *= prime(1+A003415(d)))); (m); };
%o A319357 v319357 = rgs_transform(vector(up_to,n,A319356(n)));
%o A319357 A319357(n) = v319357[n];
%Y A319357 Cf. A003415, A319356.
%Y A319357 Cf. A000041 (positions of 2's), A001248 (positions of 3's), A006881 (positions of 4's),
%Y A319357 Cf. also A300245, A300249, A319353, A319693.
%K A319357 nonn
%O A319357 1,2
%A A319357 _Antti Karttunen_, Oct 02 2018