cp's OEIS Frontend

A320004 Filter sequence combining the largest proper divisor of n (A032742) with n's residue modulo 4 (A010873), and a single bit (A319710) telling whether the smallest prime factor is unitary.

%I A320004 #16 Jan 19 2019 04:15:43
%S A320004 1,2,3,4,5,6,3,7,8,9,3,10,5,11,12,13,5,14,3,15,16,17,3,18,19,20,21,22,
%T A320004 5,23,3,24,25,26,27,28,5,29,30,31,5,32,3,33,34,35,3,36,37,38,39,40,5,
%U A320004 41,42,43,44,45,3,46,5,47,48,49,50,51,3,52,53,54,3,55,5,56,57,58,25,59,3,60,61,62,3,63,64,65,66,67,5,68,30,69,70,71,72,73,5,74,75,76,5,77,3
%N A320004 Filter sequence combining the largest proper divisor of n (A032742) with n's residue modulo 4 (A010873), and a single bit (A319710) telling whether the smallest prime factor is unitary.
%C A320004 Restricted growth sequence transform of triple [A010873(A020639(n)), A032742(n), A319710(n)], or equally, of ordered pair [A319714(n), A319710(n)].
%C A320004 Here any nontrivial equivalence classes (that is, when we exclude the singleton classes and two infinite classes of A002144 and A002145), like the example shown, may not contain any even numbers, nor any numbers from A283050. See additional comments in A319717 and A319994.
%C A320004 For all i, j:
%C A320004   a(i) = a(j) => A024362(i) = A024362(j),
%C A320004   a(i) = a(j) => A067029(i) = A067029(j),
%C A320004   a(i) = a(j) => A071178(i) = A071178(j),
%C A320004   a(i) = a(j) => A077462(i) = A077462(j) => A101296(i) = A101296(j).
%H A320004 Antti Karttunen, <a href="/A320004/b320004.txt">Table of n, a(n) for n = 1..100000</a>
%e A320004 For n = 33 (3*11) and n = 77 (7*11), the modulo 4 residue of the smallest prime factor is 3, and the largest proper divisors (A032742) is also equal 11, and the smallest prime factor is unitary. Thus a(33) = a(77) (= 25, a running count value allotted by rgs-transform).
%o A320004 (PARI)
%o A320004 up_to = 100000;
%o A320004 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 A320004 A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
%o A320004 A286474(n) = if(1==n,n,(4*A032742(n) + (n % 4)));
%o A320004 A319710(n) = ((n>1)&&(factor(n)[1,2]>1));
%o A320004 v320004 = rgs_transform(vector(up_to,n,[A286474(n),A319710(n)]));
%o A320004 A320004(n) = v320004[n];
%Y A320004 Cf. A319704, A319714, A319994.
%Y A320004 Cf. also A319717 (analogous sequence for modulo 6 residues).
%Y A320004 Cf. A002145 (positions of 3's), A002144 (positions of 5's).
%Y A320004 Differs from A319704 for the first time at n=77, and from A319714 for the first time at n=49.
%K A320004 nonn
%O A320004 1,2
%A A320004 _Antti Karttunen_, Oct 04 2018