A072499 -id:A072499 - cp's OEIS frontend

A072504 a(n) = LCM of divisors of n which are <= sqrt(n).

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 6, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 5, 2, 3, 4, 1, 30, 1, 4, 3, 2, 5, 12, 1, 2, 3, 20, 1, 6, 1, 4, 15, 2, 1, 12, 7, 10, 3, 4, 1, 6, 5, 28, 3, 2, 1, 60, 1, 2, 21, 8, 5, 6, 1, 4, 3, 70, 1, 24, 1, 2, 15, 4, 7, 6, 1, 40, 9, 2, 1, 84, 5, 2, 3, 8, 1, 90, 7, 4, 3, 2, 5, 24

Offset: 1

Amarnath Murthy, Jul 20 2002

nonn

			a(20) = 4: the divisors of 20 are 1,2,4,5,10 and 20; a(20) = lcm(1,2,4) = 4.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A072505.

Cf. A161906, A072499.

a072504 = foldl1 lcm . a161906_row  -- Reinhard Zumkeller, Mar 08 2013

A072504 := proc(n)
    local ds ;
    ds := [] ;
    for d in numtheory[divisors](n) do
        if d^2 <= n then
            ds := [op(ds),d] ;
        end if;
    end do:
    ilcm(op(ds)) ;
end proc:
seq(A072504(n),n=1..20) ; # R. J. Mathar, Oct 03 2014

Table[LCM@@Select[Divisors[n],#<=Sqrt[n]&],{n,100}] (* Harvey P. Dale, Aug 26 2014 *)

More terms from Matthew Conroy, Sep 09 2002

A072501 Ratio of the product of divisors of n which are > n^(1/2) to product of divisors of n which are < n^(1/2).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 16, 9, 25, 11, 48, 13, 49, 25, 64, 17, 162, 19, 125, 49, 121, 23, 576, 25, 169, 81, 343, 29, 900, 31, 512, 121, 289, 49, 2916, 37, 361, 169, 1600, 41, 2401, 43, 1331, 405, 529, 47, 12288, 49, 1250, 289, 2197, 53, 6561, 121, 3136, 361, 841, 59

Offset: 1

Amarnath Murthy, Jul 20 2002

nonn

It can easily be proved that the ratio is always an integer. a(n) = n if n is a prime or the square of a prime.

If 1/3 were chosen as the exponent instead of 1/2, then the sequence would begin: 1, 2, 3, 8, 5, 36, 7, 32, 27, .... If the exponent is decreased along 1/4, 1/5, ..., then the resulting sequence tends towards A007955. - Michel Marcus, Sep 17 2013

			a(20) = 25. The divisors of 20 are 1,2,4,5,10 and 20. a(20) = 10*20/2*4 = 25.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Ratio of corresponding terms of A072500 and A072499.

Table[Times @@ ((d = Divisors[n])^Sign[d - Sqrt[n]]), {n, 1, 59}] (* Ivan Neretin, May 01 2016 *)

a(n) = {d = divisors(n); pa = 1; pb = 1; fordiv(n, d, if (d^2 < n, pa *= d); if (d^2 > n, pb *= d);); pb/pa;} \\ Michel Marcus, Sep 17 2013

More terms from Sascha Kurz, Feb 02 2003

A363520 Product of the divisors of n that are < sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 8, 3, 2, 1, 24, 1, 2, 3, 8, 1, 30, 1, 8, 3, 2, 5, 24, 1, 2, 3, 40, 1, 36, 1, 8, 15, 2, 1, 144, 1, 10, 3, 8, 1, 36, 5, 56, 3, 2, 1, 720, 1, 2, 21, 8, 5, 36, 1, 8, 3, 70, 1, 1152, 1, 2, 15, 8, 7, 36, 1, 320, 3, 2, 1

Offset: 1

Wesley Ivan Hurt, Jun 07 2023

			The product of divisors of 16 that are < sqrt(16) = 4 is 1*2 = 2, so a(16) = 2.

Cf. A000196, A010052, A072499, A363521.

Cf. A070039 (sum of those divisors).

a[n_] := Times @@ Select[Divisors[n], #^2 < n &]; Array[a, 100]

a(n) = vecprod(select(x->(x^2Michel Marcus, Jun 08 2023

a(n) = Product_{d|n, d

a(n) = Product_{k=1..floor(sqrt(n-1))} k^c(n/k), where c(m) = 1-ceiling(m)+floor(m).

a(n) = A072499(n)/A000196(n)^A010052(n) for n>=1.

A363521 Product of the divisors d of n such that sqrt(n) < d < n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 4, 1, 5, 1, 24, 1, 7, 5, 8, 1, 54, 1, 50, 7, 11, 1, 576, 1, 13, 9, 98, 1, 900, 1, 128, 11, 17, 7, 1944, 1, 19, 13, 1600, 1, 2058, 1, 242, 135, 23, 1, 36864, 1, 250, 17, 338, 1, 4374, 11, 3136, 19, 29, 1, 1080000, 1, 31, 189, 512, 13, 7986, 1, 578, 23

Offset: 1

Wesley Ivan Hurt, Jun 07 2023

			The divisors of 16 are {1,2,4,8,16} and the product of the divisors d of n such that sqrt(16) = 4 < d < 16 is 8, so a(16) = 8.
The divisors of 30 are {1,2,3,5,6,10,15,30} and the product of the divisors d of n such that sqrt(30) < d < 30 is 6*10*15 = 900, so a(30) = 900.

Cf. A007955, A007956, A072499, A363520.

a[n_] := Product[If[n < d^2 < n^2, d, 1], {d, Divisors[n]}]; Array[a, 100] (* Amiram Eldar, Jun 08 2023 *)

a(n) = vecprod(select(x->((sqrt(n)Michel Marcus, Jun 08 2023

a(n) = Product_{d|n, sqrt(n) < d < n} d.
a(n) = A007956(n)/A072499(n).
a(n) = A007955(n)/(n*A072499(n)).

A379872 Numbers k that are the product of the lower half of their nontrivial divisors.

Original entry on oeis.org

1, 24, 30, 40, 56, 64, 70, 105, 135, 154, 165, 182, 189, 195, 231, 273, 286, 297, 351, 357, 374, 385, 399, 418, 429, 442, 455, 459, 494, 513, 561, 595, 598, 621, 627, 646, 663, 665, 715, 729, 741, 759, 782, 805, 874, 875, 897, 935, 957, 969, 986, 1001, 1015, 1023, 1045, 1054, 1085

Offset: 1

Tom Gadron, Jan 04 2025

nonn

All terms under 1 million except 1, 2^6, 3^6, 5^6 and 7^6  have 6 nontrivial divisors, with p^6 for p prime having 5 nontrivial divisors, and so it seems that each term in the sequence is the product of three distinct numbers. - Edited by Robert Israel, Feb 04 2025
The majority of the terms are the product of 3 primes, but there are also terms of the form p*q*p^2, p*p^2*q, or p*p^2*p^3.
The first consecutive integers that appear in the sequence are a(45)=874 and a(46)=875.

			24 is a term because the nontrivial divisors of 24 are 2,3,4,6,8,12, and 24=2*3*4.
30 is a term because the nontrivial divisors of 30 are 2,3,5,6,10,15, and 30=2*3*5.
135 is a term because the nontrivial divisors of 135 are 3,5,9,15,27,45, and 135=3*5*9.
729 is a term because the nontrivial divisors of 729 are 3,9,27,81,243, and 729=3*9*27.

Robert Israel, Table of n, a(n) for n = 1..10000
Tom Gadron, Java Program

Cf. A072499.

Java
```
\\ See Gadron link.
```

isA379872 := proc(n)
    local d;
    numtheory[divisors](n) minus {1,n} ;
    d := sort(convert(%,set)) ;
    mul( op(i,d),i=1..floor((nops(d)+1)/2)) ;
    if % = n then
        true;
    else
        false;
    end if;
end proc:
A379872 := proc(n)
    option remember ;
    if n =1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA379872(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A379872(n),n=1..60) ; # R. J. Mathar, Jan 29 2025

q[k_] := Times @@ Select[Divisors[k], #^2 <= k &] == k; Select[Range[1200], q] (* Amiram Eldar, Jan 05 2025 *)

isok(k) = my(d=divisors(k)); d=setminus(d, Set([1,k])); vecprod(Vec(d, #d\2 + #d%2)) == k; \\ Michel Marcus, Jan 05 2025

A382486 Product of distinct prime divisors of n that are <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 5, 2, 3, 2, 1, 30, 1, 2, 3, 2, 5, 6, 1, 2, 3, 10, 1, 6, 1, 2, 15, 2, 1, 6, 7, 10, 3, 2, 1, 6, 5, 14, 3, 2, 1, 30, 1, 2, 21, 2, 5, 6, 1, 2, 3, 70, 1, 6, 1, 2, 15, 2, 7, 6, 1, 10, 3, 2, 1, 42, 5

Offset: 1

Ilya Gutkovskiy, Apr 10 2025

nonn

Cf. A007947, A007955, A072499, A097974.

Table[Times @@ Select[Divisors[n], PrimeQ[#] && # <= Sqrt[n] &], {n, 1, 85}]

a(n) = vecprod(select(x->x<=sqrt(n), factor(n)[,1])); \\ Michel Marcus, Apr 17 2025

a(p) = 1, for prime p.

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A072504 a(n) = LCM of divisors of n which are <= sqrt(n).

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A072501 Ratio of the product of divisors of n which are > n^(1/2) to product of divisors of n which are < n^(1/2).

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A363520 Product of the divisors of n that are < sqrt(n).

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A363521 Product of the divisors d of n such that sqrt(n) < d < n.

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A379872 Numbers k that are the product of the lower half of their nontrivial divisors.

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A382486 Product of distinct prime divisors of n that are <= sqrt(n).

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