A068795 -id:A068795 - cp's OEIS frontend

A068794 In prime factorization of n replace all primes with the least prime factor of n; a(1)=1.

1, 2, 3, 4, 5, 4, 7, 8, 9, 4, 11, 8, 13, 4, 9, 16, 17, 8, 19, 8, 9, 4, 23, 16, 25, 4, 27, 8, 29, 8, 31, 32, 9, 4, 25, 16, 37, 4, 9, 16, 41, 8, 43, 8, 27, 4, 47, 32, 49, 8, 9, 8, 53, 16, 25, 16, 9, 4, 59, 16, 61, 4, 27, 64, 25, 8, 67, 8, 9, 8, 71, 32

Offset: 1

Reinhard Zumkeller, Mar 05 2002

nonn

a(n)<=n<=A068795(n);
a(n) = A068795(n) iff n=1 or n=p^k for some prime p, k>0.
a(30)=a(2*3*5)=2*2*2=8.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

lpf[n_]:=Module[{fi=FactorInteger[n]},Times@@PadRight[{},Total[ fi[[All,2]]], fi[[1,1]]]]; Array[lpf,80] (* Harvey P. Dale, Apr 14 2020 *)

a(n) = A020639(n)^A001222(n).

A079871 a(1)=1 and for n>1: ceiling(n^(1/Omega(n)))^Omega(n), where Omega(n) is the total number of prime factors of n (A001222).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 8, 9, 16, 11, 27, 13, 16, 16, 16, 17, 27, 19, 27, 25, 25, 23, 81, 25, 36, 27, 64, 29, 64, 31, 32, 36, 36, 36, 81, 37, 49, 49, 81, 41, 64, 43, 64, 64, 49, 47, 243, 49, 64, 64, 64, 53, 81, 64, 81, 64, 64, 59, 81, 61, 64, 64, 64, 81, 125, 67, 125, 81, 125, 71

Offset: 1

Reinhard Zumkeller, Jan 13 2003

nonn

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A001222, A006530, A020639, A068794, A068795, A079869, A079870.

A079871[n_] := If [n == 1, 1, Ceiling[n^(1/#)]^# & [PrimeOmega[n]]];
Array[A079871, 100] (* Paolo Xausa, Oct 27 2024 *)

a(n) = if (n==1, 1, ceil(n^(1/bigomega(n)))^bigomega(n)); \\ Michel Marcus, May 31 2016

a(n) = A079870(n)^A001222(n).
a(n) >= A079869(n); A020639(n) <= a(n) <= A006530(n);
a(m) = m = A079869(m) iff m is a prime power (A000961).

A079869 a(1)=1 and for n>1: round(n^(1/Omega(n)))^Omega(n), where Omega(n) is the total number of prime factors of n (A001222).

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 7, 8, 9, 9, 11, 8, 13, 16, 16, 16, 17, 27, 19, 27, 25, 25, 23, 16, 25, 25, 27, 27, 29, 27, 31, 32, 36, 36, 36, 16, 37, 36, 36, 81, 41, 27, 43, 64, 64, 49, 47, 32, 49, 64, 49, 64, 53, 81, 49, 81, 64, 64, 59, 81, 61, 64, 64, 64, 64, 64, 67, 64, 64, 64, 71, 32, 73

Offset: 1

Reinhard Zumkeller, Jan 13 2003

nonn

A079867(n)<=a(n)<=A079869(n); A020639(n)<=a(n)<=A006530(n);

a(m)=m=A079867(m)=A079871(m) iff m is a prime power (A000961).

a(n)=A079868(n)^A001222(n). Cf. A079872, A068794, A068795.

ron[n_]:=Module[{c=PrimeOmega[n]},Round[n^(1/c)]^c]; Join[{1},Array[ ron,80,2]] (* Harvey P. Dale, Jun 17 2020 *)

A079870 a(1)=1 and for n>1: ceiling(n^(1/Omega(n))), where Omega(n) is the total number of prime factors of n (A001222).

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 3, 4, 11, 3, 13, 4, 4, 2, 17, 3, 19, 3, 5, 5, 23, 3, 5, 6, 3, 4, 29, 4, 31, 2, 6, 6, 6, 3, 37, 7, 7, 3, 41, 4, 43, 4, 4, 7, 47, 3, 7, 4, 8, 4, 53, 3, 8, 3, 8, 8, 59, 3, 61, 8, 4, 2, 9, 5, 67, 5, 9, 5, 71, 3, 73, 9, 5, 5, 9, 5, 79, 3, 3, 10, 83, 4, 10, 10, 10, 4, 89, 4, 10, 5, 10

Offset: 1

Reinhard Zumkeller, Jan 13 2003

nonn

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A001222, A006530, A020639, A068794, A068795, A079869, A079871.

A079870[n_] := If [n == 1, 1, Ceiling[n^(1/PrimeOmega[n])]];
Array[A079870, 100] (* Paolo Xausa, Oct 28 2024 *)

a(n) = if (n==1, 1, ceil(n^(1/bigomega(n)))); \\ Michel Marcus, May 31 2016

A079871(n) = a(n)^A001222(n).
a(n) >= A079868(n); A020639(n) <= a(n) <= A006530(n);
a(m) = A079868(m) iff m is a prime power (A000961).

A376567 a(n) = binomial(bigomega(n) + omega(n), omega(n)), where bigomega = A001222 and omega = A001221.

Original entry on oeis.org

1, 2, 2, 3, 2, 6, 2, 4, 3, 6, 2, 10, 2, 6, 6, 5, 2, 10, 2, 10, 6, 6, 2, 15, 3, 6, 4, 10, 2, 20, 2, 6, 6, 6, 6, 15, 2, 6, 6, 15, 2, 20, 2, 10, 10, 6, 2, 21, 3, 10, 6, 10, 2, 15, 6, 15, 6, 6, 2, 35, 2, 6, 10, 7, 6, 20, 2, 10, 6, 20, 2, 21, 2, 6, 10, 10, 6, 20, 2

Offset: 1

Michael De Vlieger, Oct 09 2024

For prime power p^k, a(p^k) = A010846(p^k) = A000005(p^k) = k+1. Therefore, for prime p, a(p) = A010846(p) = A000005(p) = 2.

For n in A024619, a(n) != A010846(n) and A010846(n) > A000005(n).

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A001221, A001222, A007947, A010846, A024619, A068795, A376248, A376846, A376847.

with(NumberTheory):
a := n -> binomial(Omega(n) + Omega(n, distinct), Omega(n, distinct)):
seq(a(n), n = 1..79);  # Peter Luschny, Oct 25 2024

Array[Binomial[#2 + #1, #1] & @@ {PrimeNu[#], PrimeOmega[#]} &, 120]

a(n) = length of row n of A376248.

a(n) = A010846(n) - A376846(n) + A376847(n).

A079867 a(1)=1 and for n>1: floor(n^(1/Omega(n)))^Omega(n), where Omega(n) is the total number of prime factors of n (A001222).

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 7, 8, 9, 9, 11, 8, 13, 9, 9, 16, 17, 8, 19, 8, 16, 16, 23, 16, 25, 25, 27, 27, 29, 27, 31, 32, 25, 25, 25, 16, 37, 36, 36, 16, 41, 27, 43, 27, 27, 36, 47, 32, 49, 27, 49, 27, 53, 16, 49, 16, 49, 49, 59, 16, 61, 49, 27, 64, 64, 64, 67, 64, 64, 64, 71, 32, 73, 64

Offset: 1

Reinhard Zumkeller, Jan 13 2003

nonn

a(n)<=A079869(n); A020639(n)<=a(n)<=A006530(n);

a(m)=m=A079869(m)=A079871(m) iff m is a prime power (A000961).

a(n)=A079866(n)^A001222(n), cf. A068794, A068795.

Join[{1},Table[Floor[n^(1/PrimeOmega[n])]^PrimeOmega[n],{n,2,80}]] (* Harvey P. Dale, May 19 2018 *)

A079881 mpf(n)^Omega(n), where mpf(n) is the median prime factor of n (A079879).

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 7, 8, 9, 4, 11, 8, 13, 4, 9, 16, 17, 27, 19, 8, 9, 4, 23, 16, 25, 4, 27, 8, 29, 27, 31, 32, 9, 4, 25, 16, 37, 4, 9, 16, 41, 27, 43, 8, 27, 4, 47, 32, 49, 125, 9, 8, 53, 81, 25, 16, 9, 4, 59, 16, 61, 4, 27, 64, 25, 27, 67, 8, 9, 125, 71, 32, 73, 4, 125, 8, 49, 27, 79, 32

Offset: 1

Reinhard Zumkeller, Jan 13 2003

nonn

A068794(n)<=a(n)<=A068795(n);

a(m)=m=A068794(m)=A068795(m) iff m is a prime power (A000961).

a(n)=A079879(n)^A001222(n), A033676, O000004.

A376847 Number of m > n such that rad(m) | n and Omega(m) <= Omega(n), where rad = A007947 and Omega = A001222.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 1, 0, 3, 1, 1, 0, 4, 0, 1, 0, 3, 0, 4, 0, 0, 1, 1, 1, 2, 0, 1, 1, 5, 0, 5, 0, 3, 2, 1, 0, 6, 0, 1, 1, 3, 0, 1, 1, 5, 1, 1, 0, 11, 0, 1, 2, 0, 1, 5, 0, 3, 1, 5, 0, 4, 0, 1, 1, 3, 1, 5, 0, 8, 0, 1, 0, 11, 1, 1, 1

Offset: 1

Michael De Vlieger, Oct 13 2024

nonn

			Table of select n such that a(n) > 0:
   n  a(n)  List of m in A376248 such that Omega(m) <= Omega(n)
  -------------------------------------------------------------
   6    1   {9}
  10    1   {25}
  12    2   {18, 27}
  14    1   {49}
  15    1   {25}
  18    1   {27}
  20    3   {25, 50, 125}
  24    4   {27, 36, 54, 81}
  28    3   {49, 98, 343}
  30    4   {45, 50, 75, 125}
  40    5   {50, 100, 125, 250, 625}
  48    6   {54, 72, 81, 108, 162, 243}
  60   11   {75, 81, 90, 100, 125, 135, 150, 225, 250, 375, 625}

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Hasse diagrams of m in select R(n), where R(n) is the union of rows n of A162306 and A376248, indicating in blue those m > n such that Omega(m) <= Omega(n).

Cf. A000961, A001221, A001222, A007947, A162306, A376567, A376846, A068795.

with(NumberTheory):
cond := (m, n) -> irem(n, Radical(m)) = 0 and Omega(m) <= Omega(n):
a := n -> nops(select(m -> cond(m, n), [seq(n+1..A068795(n))])):
seq(a(n), n = 1..87);  # Peter Luschny, Oct 25 2024

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
Table[k = PrimeOmega[n]; w = PrimeNu[n]; Binomial[k + w, w] - Count[Range[n], _?(And[Divisible[n, rad[#]], PrimeOmega[#] > k] &)], {n, 120}]

a(n) = card({m > n : rad(m) | n, Omega(m) <= Omega(n) }).
a(n) = 0 for prime power n (in A000961).
a(n) = card(A376248 \ A162306).
a(n) = A376567(n) - A010846(n) + A376546(n) = binomial(A001222(n) + A001221(n), A001221(n)) - A010846(n) + A376546(n).

cp's OEIS Frontend

A068794 In prime factorization of n replace all primes with the least prime factor of n; a(1)=1.

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A079871 a(1)=1 and for n>1: ceiling(n^(1/Omega(n)))^Omega(n), where Omega(n) is the total number of prime factors of n (A001222).

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A079869 a(1)=1 and for n>1: round(n^(1/Omega(n)))^Omega(n), where Omega(n) is the total number of prime factors of n (A001222).

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A079870 a(1)=1 and for n>1: ceiling(n^(1/Omega(n))), where Omega(n) is the total number of prime factors of n (A001222).

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A376567 a(n) = binomial(bigomega(n) + omega(n), omega(n)), where bigomega = A001222 and omega = A001221.

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A079867 a(1)=1 and for n>1: floor(n^(1/Omega(n)))^Omega(n), where Omega(n) is the total number of prime factors of n (A001222).

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A079881 mpf(n)^Omega(n), where mpf(n) is the median prime factor of n (A079879).

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A376847 Number of m > n such that rad(m) | n and Omega(m) <= Omega(n), where rad = A007947 and Omega = A001222.

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