A327931 - OEIS

%I #20 Sep 30 2019 20:20:17

%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 28,40,41,42,2,43,2,44,45,46,47,48,2,49,50,51,2,52,2,53,54,55,47,56,2,57,58,59,2,60,41,61,62,63,2,64,37,65,66,67,68,69,2,70,71,72,2,73,2,74,75

%N Lexicographically earliest infinite sequence such that for all i, j, a(i) = a(j) => A327930(i) = A327930(j).

%C Restricted growth sequence transform of A327930, or equally, of the ordered pair [A003415(n), A319356(n)].

%C It seems that the sequence takes duplicated values only on primes (A000040) and some subset of squarefree semiprimes (A006881). If this holds, then also the last implication given below is valid.

%C For all i, j:

%C   a(i) = a(j) => A000005(i) = A000005(j),

%C   a(i) = a(j) => A319684(i) = A319684(j),

%C   a(i) = a(j) => A319685(i) = A319685(j),

%C   a(i) = a(j) => A101296(i) = A101296(j). [Conjectural, see notes above and in A319357]

%H Antti Karttunen, <a href="/A327931/b327931.txt">Table of n, a(n) for n = 1..65537</a>

%F a(p) = 2 for all primes p.

%e Divisors of 39 are [1, 3, 13, 39], while the divisors of 55 are [1, 5, 11, 55]. Taking their arithmetic derivatives (A003415) yields in both cases [0, 1, 1, 16], thus a(39) = a(55) (= 28, as allotted by restricted growth sequence transform).

%o (PARI)

%o up_to = 8192;

%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 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 v003415 = vector(up_to,n,A003415(n));

%o A327930(n) = { my(m=1); fordiv(n,d,if((d>1), m *= prime(v003415[d]))); (m); };

%o v327931 = rgs_transform(vector(up_to, n, A327930(n)));

%o A327931(n) = v327931[n];

%Y Cf. A000005, A000040 (positions of 2's), A003415, A006881, A101296, A319356, A319357, A319684, A319685, A327930.

%Y Differs from A300249 for the first time at n=105, where a(105)=75, while A300249(105)=56.

%Y Differs from A300235 for the first time at n=153, where a(153)=110, while A300235(153)=106.

%Y Differs from A305895 for the first time at n=3283, where a(3283)=2502, while A305895(3283)=1845.

%K nonn

%O 1,2

%A _Antti Karttunen_, Sep 30 2019