A129128 -id:A129128 - cp's OEIS frontend

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

Antti Karttunen, Dec 20 2014

nonn

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, ...

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

Variants: A252462, A252463.
Bisections: A000027, A129128.
Cf. A001222, A008578, A032742, A055396, A056239, A064989.

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 *)

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

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.

A129127 An irregular table of node numbers (in order of appearance) which generate two children in the tree described by A128628 and A129129.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 14, 15, 18, 13, 22, 21, 30, 17, 26, 33, 35, 42, 50, 54, 19, 34, 39, 55, 66, 70, 75, 90

Offset: 1

Alford Arnold, Apr 01 2007

Shape sequence for A129127 is A002865 since the odd numbers can be mapped to cyclic partitions. The sequence contains the same values as A129128 which is sorted by the associated odd numbers.

Cf. A002865, A128628, A129128, A129129.

cp's OEIS Frontend

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.

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