A252461 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.

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

Offset: 1

Author

Antti Karttunen, Dec 20 2014

Keywords

• nonn

Comments

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

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

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Variants: A252462, A252463.

Bisections: A000027, A129128.

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

Programs

Mathematica

a252461[n_Integer] := Block[{a008578, a032742, a055396, a},
  a008578[x_] := If[x == 1, 1, Prime[x - 1]];
  a032742[x_] := If[x == 1, 1, Divisors[x][[-2]]];
  a055396[x_] := PrimePi[FactorInteger[x][[1]][[1]]];
  a[1] = 1;
  a[x_] := a008578[a055396[x]]*a032742[x];
Array[a, n]]; a252461[84] (* Michael De Vlieger, Dec 21 2014 *)

Scheme

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

Formula

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

Other identities. For all n >= 1:

a(2n) = n.

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

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