A252462 Shift one instance of the largest prime one step towards smaller primes: a(1) = 1, a(2^n) = 2^(n-1), and for other numbers: a(n) = (n / prime(g)) * prime(g-1), where g = A061395(n), index of the greatest prime dividing n.

1, 1, 2, 2, 3, 4, 5, 4, 6, 6, 7, 8, 11, 10, 9, 8, 13, 12, 17, 12, 15, 14, 19, 16, 15, 22, 18, 20, 23, 18, 29, 16, 21, 26, 25, 24, 31, 34, 33, 24, 37, 30, 41, 28, 27, 38, 43, 32, 35, 30, 39, 44, 47, 36, 35, 40, 51, 46, 53, 36, 59, 58, 45, 32, 55, 42, 61, 52, 57, 50, 67, 48, 71, 62, 45, 68, 49, 66, 73, 48, 54, 74, 79, 60

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.

Links

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

Crossrefs

Variant: A252461.

Cf. A001222, A008578, A052126, A061395, A056239, A064989, A209229, A252735.

Programs

Mathematica

a252462[n_Integer] := Block[{a008578, a052126, a061395, a},
  a008578[x_] := If[x == 1, 1, Prime[x - 1]];
  a052126[x_] := If[x == 1, 1, x/FactorInteger[x][[-1]][[1]]];
  a061395[x_] := PrimePi[FactorInteger[x][[-1]][[1]]];
  a[1] = 1;
  a[x_] := a008578[a061395[x]]*a052126[x];
Array[a, n]]; a252462[84] (* Michael De Vlieger, Dec 21 2014 *)

Scheme

(define (A252462 n) (if (= 1 n) n  (* (A008578 (A061395 n)) (A052126 n))))

Formula

a(1) = 1; for n>1: a(n) = A008578(A061395(n)) * A052126(n). [Compare to the similar formula given for A064989.]

a(n) = A008578(1 + A252735(n)) * A052126(n).

Other identities. For all n >= 1:

a(2^n) = 2^(n-1).

For n >= 2, A001222(a(n)) = A001222(n) - A209229(n). [Number of prime divisors decreases only when n is a power of 2 larger than 1.]