cp's OEIS Frontend

A284003 a(n) = A007913(A283477(n)) = A019565(A006068(n)).

%I A284003 #39 Apr 27 2018 17:10:30
%S A284003 1,2,6,3,30,15,5,10,210,105,35,70,7,14,42,21,2310,1155,385,770,77,154,
%T A284003 462,231,11,22,66,33,330,165,55,110,30030,15015,5005,10010,1001,2002,
%U A284003 6006,3003,143,286,858,429,4290,2145,715,1430,13,26,78,39,390,195,65,130,2730,1365,455,910,91,182,546,273,510510,255255,85085,170170,17017
%N A284003 a(n) = A007913(A283477(n)) = A019565(A006068(n)).
%C A284003 A squarefree analog of A302783. Each term is either a divisor or a multiple of the next one. In contrast to A302033 at each step the previous term can be multiplied (or divided), not just by a single prime, but possibly by a product of several distinct ones, A019565(A000975(k)). E.g., a(3) = 3, a(4) = 2*5*a(3) = 30. - _Antti Karttunen_, Apr 17 2018
%H A284003 Antti Karttunen, <a href="/A284003/b284003.txt">Table of n, a(n) for n = 0..8191</a>
%H A284003 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A284003 a(n) = A007913(A283477(n)).
%F A284003 Other identities. For all n >= 0:
%F A284003 A048675(a(n)) = A006068(n).
%F A284003 A046523(a(n)) = A284004(n).
%F A284003 It seems that A001222(a(n)) = A209281(n).
%F A284003 a(n) = A019565(A006068(n)) = A302033(A064707(n)). - _Antti Karttunen_, Apr 16 2018
%t A284003 Table[Apply[Times, FactorInteger[#] /. {p_, e_} /; e > 0 :> Times @@ (p^Mod[e, 2])] &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e == 1 :> {Times @@ Prime@ Range@ PrimePi@ p, e}] &[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2]]], {n, 0, 52}] (* _Michael De Vlieger_, Mar 18 2017 *)
%o A284003 (PARI)
%o A284003 A007913(n) = core(n);
%o A284003 A034386(n) = prod(i=1, primepi(n), prime(i));
%o A284003 A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From _Charles R Greathouse IV_, Jun 28 2015
%o A284003 A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from _M. F. Hasler_
%o A284003 A283477(n) = A108951(A019565(n));
%o A284003 A284003(n) = A007913(A283477(n));
%o A284003 (PARI)
%o A284003 A006068(n)= { my(s=1, ns); while(1, ns = n >> s; if(0==ns, break()); n = bitxor(n, ns); s <<= 1; ); return (n); } \\ From A006068
%o A284003 A284003(n) = A019565(A006068(n)); \\ (and use A019565 from above) - _Antti Karttunen_, Apr 16 2018
%o A284003 (Scheme) (define (A284003 n) (A007913 (A283477 n)))
%Y A284003 A permutation of A005117.
%Y A284003 Cf. A000975, A001222, A006068, A007913, A019565, A046523, A048675, A064707, A209281, A283477, A284004.
%Y A284003 Cf. also A277811, A283475, A302033, A302783.
%K A284003 nonn,look
%O A284003 0,2
%A A284003 _Antti Karttunen_, Mar 18 2017
%E A284003 Name amended with a second formula by _Antti Karttunen_, Apr 16 2018