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

1,3

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

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.

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 *)

Prog

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

Crossrefs

Variants: A252462, A252463.

Bisections: A000027, A129128.

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

Sequence in context: A007150 A213634 A300271 * A323608 A122352 A338903

Adjacent sequences: A252458 A252459 A252460 * A252462 A252463 A252464

Keyword

nonn

Author

Antti Karttunen, Dec 20 2014

Status

approved