A361566 -id:A361566 - cp's OEIS frontend

A361565 a(n) is the numerator of the median of divisors of n.

1, 3, 2, 2, 3, 5, 4, 3, 3, 7, 6, 7, 7, 9, 4, 4, 9, 9, 10, 9, 5, 13, 12, 5, 5, 15, 6, 11, 15, 11, 16, 6, 7, 19, 6, 6, 19, 21, 8, 13, 21, 13, 22, 15, 7, 25, 24, 7, 7, 15, 10, 17, 27, 15, 8, 15, 11, 31, 30, 8, 31, 33, 8, 8, 9, 17, 34, 21, 13, 17, 36, 17, 37, 39, 10

Offset: 1

Stefano Spezia, Mar 15 2023

			a(9) = 3 since the divisors of 9 are 1, 3, 9, and their median is 3.
a(12) = 7 since the divisors of 12 are 1, 2, 3, 4, 6, 12, and their median is 7/2.

Winston de Greef, Table of n, a(n) for n = 1..10000
Index entries for sequences related to divisors of numbers

Cf. A027750, A033676, A033677, A071090, A139710, A139711, A361566 (denominator).

a[n_]:=Numerator[Median[Divisors[n]]]; Array[a,75]

a(n) = my(d=divisors(n), m=#d+1); numerator((d[m\2] + d[m-m\2])/2); \\ Michel Marcus, Mar 16 2023

a(n) = numerator((A033676(n) + A033677(n))/2).
a(p) = (1 + p)/2 if p is an odd prime.
For p a prime, a(p^k) = (p^((k-1)/2) + p^((k+1)/2))/2 if k is odd, a(p^k) = p^(k/2) if k is even.

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

A361631 a(n) is the denominator of the median of the distinct prime factors of n.

Original entry on oeis.org

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

Offset: 2

Stefano Spezia, Mar 18 2023

			a(12) = 2 since the distinct prime factors of 12 are 2 and 3, of median equal to 5/2.
a(30) = 1 since the distinct prime factors of 30 are 2, 3, and 5, of median equal to 3.

Winston de Greef, Table of n, a(n) for n = 2..10000

Cf. A001221, A027748, A323172, A361566, A361630 (numerator), A361633 (with multiplicity).

a[n_]:=Denominator[Median[FactorInteger[n][[All, 1]]]]; Array[a,88,2]

a(n)=my(f=factor(n)[,1]~, i=length(f)); denominator(if(i%2, f[i\2+1], (f[i/2]+f[i/2+1])/2)) \\ Winston de Greef, Mar 23 2023

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

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A361565 a(n) is the numerator of the median of divisors of n.

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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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A361631 a(n) is the denominator of the median of the distinct prime factors of n.

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Author

Keywords

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Crossrefs

Programs

Mathematica

PARI

Formula