A075423 -id:A075423 - cp's OEIS frontend

A075424 a(n) = A075423(A075423(n)).

0, 1, 0, 1, 4, 5, 0, 1, 2, 9, 4, 5, 12, 13, 0, 1, 4, 5, 2, 9, 20, 21, 4, 1, 4, 1, 12, 13, 28, 29, 0, 1, 32, 33, 4, 5, 36, 37, 2, 9, 40, 41, 20, 13, 14, 45, 4, 5, 2, 9, 4, 25, 4, 5, 12, 13, 56, 57, 28, 29, 60, 9, 0, 1, 64, 65, 32, 33, 68, 69, 4, 5, 72, 13, 36, 37, 76, 77, 2, 1, 2, 81, 40, 41

Offset: 2

Reinhard Zumkeller, Sep 15 2002

From Robert Israel, Apr 11 2019: (Start)
a(n) == n (mod 2).
a(n) = 0 iff n is in A000079.
a(n) = n-2 iff n-1 is in A007674. (End)

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A000079, A007674, A075423, A075425.

A075423:= n -> convert(numtheory:-factorset(n),`*`)-1:
map(A075423@@2, [$2..100]); # Robert Israel, Apr 11 2019

A075425 Number of steps to reach 1 starting with n and iterating the map n ->rad(n)-1, where rad(n) is the squarefree kernel of n (A007947).

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 3, 4, 5, 6, 1, 2, 3, 4, 3, 4, 5, 6, 3, 2, 3, 2, 5, 6, 7, 8, 1, 2, 3, 4, 3, 4, 5, 6, 3, 4, 5, 6, 5, 6, 7, 8, 3, 4, 3, 4, 3, 4, 3, 4, 5, 6, 7, 8, 7, 8, 9, 4, 1, 2, 3, 4, 3, 4, 5, 6, 3, 4, 5, 6, 5, 6, 7, 8, 3, 2, 3, 4, 5, 6, 7, 8, 5, 6, 7, 8, 7, 8, 9, 10, 3, 4, 5, 2, 3, 4, 5, 6, 3

Offset: 1

Reinhard Zumkeller, Sep 15 2002

nonn

Sequence is defined for all n, as A075423(n) < n.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

a075425 n = snd $ until ((== 1) . fst)
                        (\(x, i) -> (a075423 x, i + 1)) (n, 0)
-- Reinhard Zumkeller, Aug 14 2013

rad(n)=vecprod(factor(n)[,1])
a(n)=my(k);while(n>1,n=rad(n)-1;k++); k \\ Charles R Greathouse IV, Jul 09 2013

A110157 a(n) = a(rad(n) - 1) + 1, where rad(n) is the squarefree kernel of n, rad=A007947.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 4, 5, 2, 3, 4, 5, 4, 5, 6, 7, 2, 3, 4, 5, 4, 5, 6, 7, 4, 3, 4, 3, 6, 7, 8, 9, 2, 3, 4, 5, 4, 5, 6, 7, 4, 5, 6, 7, 6, 7, 8, 9, 4, 5, 4, 5, 4, 5, 4, 5, 6, 7, 8, 9, 8, 9, 10, 5, 2, 3, 4, 5, 4, 5, 6, 7, 4, 5, 6, 7, 6, 7, 8, 9, 4, 3, 4, 5, 6, 7, 8, 9, 6, 7, 8, 9, 8, 9, 10, 11, 4, 5, 6, 3, 4, 5, 6, 7

Offset: 0

Reinhard Zumkeller, Jul 14 2005

nonn

a(n) = a(A075423(n)) + 1;

a(A075426(n)) = n+1 and a(m) < n+1 for m < A075426(n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

a110157 n = genericIndex a110157_list (n-1)
a110157_list = 0 : map ((+ 1) . a110157 . (+ 1)) a075423_list
-- Reinhard Zumkeller, Aug 14 2013

rad(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])
a(n)=if(n<4,n,1+a(rad(n)-1)) \\ Charles R Greathouse IV, Aug 08 2013

a(n) < 4 log_2(n) for n > 1. - Charles R Greathouse IV, Aug 08 2013

A350661 a(1) = 1; a(n) = a(A007947(n) - 1) + n.

Original entry on oeis.org

1, 3, 6, 5, 10, 16, 23, 9, 12, 22, 33, 22, 35, 49, 64, 17, 34, 28, 47, 32, 53, 75, 98, 34, 30, 56, 30, 63, 92, 122, 153, 33, 66, 100, 135, 46, 83, 121, 160, 52, 93, 135, 178, 97, 94, 140, 187, 58, 65, 62, 113, 82, 135, 64, 119, 91, 148, 206, 265, 152, 213, 275, 95, 65

Offset: 1

Gleb Ivanov, Jan 10 2022

nonn

			a(3) = a(2) + 3 = a(1) + 5 = 6;
a(6) = a(5) + 6 = a(4) + 11 = a(1) + 15 = 16.

Cf. A007947, A075423, A075425, A075424, A005187.

rad[n_] := Times @@ First /@ FactorInteger[n]; a[1] = 1; a[n_] := a[n] = a[rad[n] - 1] + n; Array[a, 100] (* Amiram Eldar, Jan 10 2022 *)

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
a(n) = if (n==1, 1, a(rad(n) - 1) + n); \\ Michel Marcus, Jan 10 2022

from sympy import prod, primefactors
from functools import lru_cache
rad = lambda n: prod(primefactors(n))
@lru_cache()
def a(n):
    if n == 1: return 1
    return a(rad(n)-1)+n
print([a(i) for i in range(1, 100)])

a(1) = 1; a(n) = a(A075423(n)) + n.

cp's OEIS Frontend

A075424 a(n) = A075423(A075423(n)).

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A075425 Number of steps to reach 1 starting with n and iterating the map n ->rad(n)-1, where rad(n) is the squarefree kernel of n (A007947).

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A110157 a(n) = a(rad(n) - 1) + 1, where rad(n) is the squarefree kernel of n, rad=A007947.

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Haskell

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A350661 a(1) = 1; a(n) = a(A007947(n) - 1) + n.

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