A252461 - OEIS

%I #23 Dec 23 2014 00:36:28

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

%T 23,15,29,16,22,17,21,18,31,19,26,20,37,21,41,22,30,23,43,24,35,25,34,

%U 26,47,27,33,28,38,29,53,30,59,31,42,32,39,33,61,34,46,35,67,36,71,37,50,38,55,39,73,40,54,41,79,42

%N Shift one instance of the smallest prime one step towards smaller primes: a(1) = 1, a(2n) = n, and for odd numbers > 1: a(n) = (n / prime(s)) * prime(s-1), where s = A055396(n), index of the smallest prime dividing n.

%C Iterating from any n as a(n), a(a(n)), a(a(a(n))), etc. reaches 1 after A056239(n) iterations.

%C Even bisection gives the natural numbers A000027, the odd bisection from the third term onward is A129128: 2, 3, 5, 6, 7, 11, 10, ...

%H Antti Karttunen, <a href="/A252461/b252461.txt">Table of n, a(n) for n = 1..10000</a>

%F a(1) = 1; for n>1: a(n) = A008578(A055396(n)) * A032742(n). [Compare to the similar formula of A064989.]

%F Other identities. For all n >= 1:

%F a(2n) = n.

%F If n is odd, A001222(a(n)) = A001222(n).

%F If n is even, A001222(a(n)) = A001222(n) - 1.

%t a252461[n_Integer] := Block[{a008578, a032742, a055396, a},

%t   a008578[x_] := If[x == 1, 1, Prime[x - 1]];

%t   a032742[x_] := If[x == 1, 1, Divisors[x][[-2]]];

%t   a055396[x_] := PrimePi[FactorInteger[x][[1]][[1]]];

%t   a[1] = 1;

%t   a[x_] := a008578[a055396[x]]*a032742[x];

%t Array[a, n]]; a252461[84] (* _Michael De Vlieger_, Dec 21 2014 *)

%o (Scheme) (define (A252461 n) (if (= 1 n) n (* (A008578 (A055396 n)) (A032742 n))))

%Y Variants: A252462, A252463.

%Y Bisections: A000027, A129128.

%Y Cf. A001222, A008578, A032742, A055396, A056239, A064989.

%K nonn

%O 1,3

%A _Antti Karttunen_, Dec 20 2014