A079879 -id:A079879 - cp's OEIS frontend

A079880 a(n) = n/mpf(n), where mpf(n) is the median prime factor of n (A079879).

1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6, 1, 7, 5, 8, 1, 6, 1, 10, 7, 11, 1, 12, 5, 13, 9, 14, 1, 10, 1, 16, 11, 17, 7, 18, 1, 19, 13, 20, 1, 14, 1, 22, 15, 23, 1, 24, 7, 10, 17, 26, 1, 18, 11, 28, 19, 29, 1, 30, 1, 31, 21, 32, 13, 22, 1, 34, 23, 14, 1, 36, 1, 37, 15, 38, 11, 26, 1, 40, 27, 41, 1, 42

Offset: 1

Reinhard Zumkeller, Jan 13 2003

A052126(n)<=a(n)<=A032742(n);

a(m)=A032742(m)=A052126(m) iff m is a prime power (A000961).

Robert Israel, Table of n, a(n) for n = 1..10000

a(n)=n/A079879(n), A033676.

f:= proc(n) local F, F2, m, i;
    F:= sort(ifactors(n)[2], (i, j) -> i[1]=F2[-1] then return n/F[i][1] fi
    od
end proc:
1, seq(f(n),n=2..100); # Robert Israel, Jan 26 2018

mpf[n_] := Module[{fi = FactorInteger[n], ff, Om}, ff = Flatten[Table[ Table[f[[1]], {f[[2]]}], {f, fi}]]; Om = Length[ff]; If[OddQ[Om], ff[[Floor[Om/2]+1]], ff[[Om/2]]]];
a[n_] := n/mpf[n];
Array[a, 100] (* Jean-François Alcover, Mar 09 2019 *)

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.

A361632 a(n) is the numerator of the median of the prime factors of n with repetition.

Original entry on oeis.org

2, 3, 2, 5, 5, 7, 2, 3, 7, 11, 2, 13, 9, 4, 2, 17, 3, 19, 2, 5, 13, 23, 2, 5, 15, 3, 2, 29, 3, 31, 2, 7, 19, 6, 5, 37, 21, 8, 2, 41, 3, 43, 2, 3, 25, 47, 2, 7, 5, 10, 2, 53, 3, 8, 2, 11, 31, 59, 5, 61, 33, 3, 2, 9, 3, 67, 2, 13, 5, 71, 2, 73, 39, 5, 2, 9, 3, 79

Offset: 2

Stefano Spezia, Mar 18 2023

			a(12) = 2 since 12 = 2*2*3, and the median of the factors is equal to 2.
a(36) = 5 since 30 = 2*2*3*3, and the median of the factors is equal to 5/2.

Cf. A001222, A027746, A079879, A323171, A361565, A361630 (without repetition), A361633 (denominator), A361725.

a[n_]:=Numerator[Median[Flatten[Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]]]]; Array[a,78,2]

For p a prime, a(p^k) = p.

a(n) = numerator((A079879(n) + A361725(n))/2).

A361633 a(n) is the denominator of the median of the prime factors of n with repetition.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1

Offset: 2

Stefano Spezia, Mar 18 2023

			a(12) = 1 since 12 = 2*2*3, and the median of the factors is equal to 2/1.
a(36) = 2 since 30 = 2*2*3*3, and the median of the factors is equal to 5/2.

Cf. A001222, A027746, A079879, A323172, A361566, A361631 (without repetition), A361632 (numerator), A361725.

a[n_]:=Denominator[Median[Flatten[ Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]]]]; Array[a,88,2]

For p a prime, a(p^k) = 1.

a(n) = denominator((A079879(n) + A361725(n))/2).

Example corrected by Peter Munn, Aug 04 2024

A361725 a(n) is the largest of two middle prime factors of n if the number of prime divisors counted with multiplicity (A001222(n)) is even, otherwise is the middle prime factor of n.

Original entry on oeis.org

2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 2, 13, 7, 5, 2, 17, 3, 19, 2, 7, 11, 23, 2, 5, 13, 3, 2, 29, 3, 31, 2, 11, 17, 7, 3, 37, 19, 13, 2, 41, 3, 43, 2, 3, 23, 47, 2, 7, 5, 17, 2, 53, 3, 11, 2, 19, 29, 59, 3, 61, 31, 3, 2, 13, 3, 67, 2, 23, 5, 71, 2, 73, 37, 5, 2, 11, 3

Offset: 2

Stefano Spezia, Mar 22 2023

nonn

			a(30) = a(2*3*5) = 3; a(60) = a(2*2*3*5) = 3; a(72) = a(2*2*2*3*3) = 2.

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

Cf. A001222, A027746, A079879, A361632, A361633.

f:= proc(n) local F,m;
  F:= sort(map(t -> t[1]$t[2],ifactors(n)[2]));
  m:= ceil((1+nops(F))/2);
  F[m]
end proc:
map(f, [$2..100]); # Robert Israel, May 19 2025

f[n_] := Block[{p = Flatten[Table[#1, {#2}] & @@@ FactorInteger@ n], len}, len = Length@ p; If[OddQ@ len, p[[(1 + len)/2]], p[[len/2+1]]]]; Table[f@ n, {n, 2, 78}] (* After Michael De Vlieger in A079879 *)

a(n) = A027746(n, floor(A001222(n)/2) + 1).
a(n) = 2*A361632(n)/A361633(n) - A079879(n) if A001222(n) is even.
a(n) = A361632(n) if A001222(n) is odd.

A261564 a(1)=2; thereafter a(n) = mpf(1+Product_{k=1..n-1} a(k)), where mpf(n) = f-th prime factor with multiplicity of n, for f=ceiling(bigomega(n)/2).

Original entry on oeis.org

2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 25621, 10300271, 2731, 1079927141307582051252331702244209088763871

Offset: 1

Anders Hellström, Aug 24 2015

Anders Hellström, Ruby program

Cf. A079879, A001222, A000945, A000946.

f[n_] := Block[{p = Flatten[Table[#1, {#2}] & @@@ FactorInteger@ n], len}, len = Length@ p; Which[n == 1, 1, OddQ@len, p[[1 + Floor[len/2]]], True, p[[len/2]]]]; a = {2}; Do[AppendTo[a, f[1 + Product[a[[k]], {k, 1, n - 1}]]], {n, 2, 13}] ; a (* Michael De Vlieger, Aug 25 2015 *)

factorsm(n)=my(v=factor(n), f=factor(n)[, 1]~, w=[]); for(i=1, #f, for(j=1, v[i, 2], w=concat(w, f[i]))); w;
mpf(n)=my(f=factorsm(n)); f[ceil(#f/2)]
first(m)=my(v=vector(m)); v[1]=2; for(i=2, m, v[i]=mpf(1+prod(j=1, i-1, v[j]))); v;

a(1)=2; thereafter a(n) = A079879(1+Product_{k=1..n-1}a(k)).

a(13) from Michael De Vlieger, Aug 25 2015

cp's OEIS Frontend

A079880 a(n) = n/mpf(n), where mpf(n) is the median prime factor of n (A079879).

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

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A361632 a(n) is the numerator of the median of the prime factors of n with repetition.

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A361633 a(n) is the denominator of the median of the prime factors of n with repetition.

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A361725 a(n) is the largest of two middle prime factors of n if the number of prime divisors counted with multiplicity (A001222(n)) is even, otherwise is the middle prime factor of n.

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A261564 a(1)=2; thereafter a(n) = mpf(1+Product_{k=1..n-1} a(k)), where mpf(n) = f-th prime factor with multiplicity of n, for f=ceiling(bigomega(n)/2).

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