A328477 - OEIS

%I #15 Oct 19 2019 21:27:56

%S 1,2,3,4,5,6,7,4,8,9,10,11,12,6,10,11,13,14,15,16,17,18,19,20,21,22,

%T 23,24,25,26,27,4,8,9,10,11,28,9,29,30,31,32,33,11,31,32,34,35,36,18,

%U 37,38,39,40,41,24,42,43,44,45,46,6,10,11,13,14,33,11,31,32,34,35,47,14,34,35,48,49,50,20,39,40,51,52,53,26,44,45,54,55,56,16,17,18,19,20,36,18

%N Lexicographically earliest infinite sequence such that a(i) = a(j) => A328469(A276086(i)) = A328469(A276086(j)) for all i, j.

%C Restricted growth sequence transform of function f(n) = A328469(A276086(n)), or equally, of the function g(0) = 0; n > 0, g(n) = [A053669(n), A278226(n)], where in the ordered pair A053669(n) gives the smallest prime not dividing n, while A278226(n) gives the prime signature of A276086(n), i.e., a signature of the multiset of nonzero digits in the primorial base expansion of n. Note that A000720(A053669(n)) = A055396(A276086(n)) is one more than the number of trailing zeros in the primorial base expansion for n > 0.

%H Antti Karttunen, <a href="/A328477/b328477.txt">Table of n, a(n) for n = 0..32768</a>

%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>

%e When written in primorial base (A049345), numbers 42 ("1200" as 42 = 1*A002110(3) + 2*A002110(2) + 0*A002110(1) + 0*A002110(0) = 1*30 + 2*6 + 0*2 + 0*1), 66 ("2100" as 66 = 2*30 + 1*6 + 0*2 + 0*1) and 222 ("10200" as 222 = 1*210 + 0*30 + 2*6 + 0*2 + 0*1) all have {1, 2} as their multiset of nonzero digits, and all have exactly two trailing zeros, thus they get an equal value in this sequence, namely a(42) = a(66) = a(222) = 33, where 33 is a running number allotted by the restricted growth sequence transform.

%o (PARI)

%o up_to = 32768;

%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 A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1);

%o A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523

%o A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };

%o Aux328469(n) = [A020639(n), A046523(n)];

%o Aux328477(n) = Aux328469(A276086(n));

%o v328477 = rgs_transform(vector(1+up_to, n, Aux328477(n-1)));

%o A328477(n) = v328477[1+n];

%Y Cf. A020639, A046523, A049345, A053669, A276086, A278226, A286626, A328399, A328469.

%K nonn

%O 0,2

%A _Antti Karttunen_, Oct 19 2019