A072504 -id:A072504 - cp's OEIS frontend

A272533 Numbers n such that A072499(n)!=A072504(n).

16, 20, 24, 28, 32, 36, 40, 42, 44, 48, 52, 54, 56, 60, 64, 66, 68, 72, 76, 78, 80, 81, 84, 88, 90, 92, 96, 99, 100, 102, 104, 108, 110, 112, 114, 116, 117, 120, 124, 126, 128, 130, 132, 135, 136, 138, 140, 144, 148, 150, 152, 153, 156, 160, 162, 164, 168, 170, 171

Offset: 1

Benedict W. J. Irwin, May 02 2016

nonn

			A072499(16) = 8 != A072504(16) = 4.

Benedict W. J. Irwin, Table of n, a(n) for n = 1..5600

Cf. A072499, A072504

isok(n) = { my(dd = select(x->x <= sqrt(n), divisors(n))); lcm(dd) != prod(k=1, #dd, dd[k]);} \\ Michel Marcus, May 06 2016

A161906 Triangle read by rows in which row n lists the divisors of n that are <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 3, 1, 2, 4, 1, 1, 2, 3, 1, 1, 2, 4, 1, 3, 1, 2, 1, 1, 2, 3, 4, 1, 5, 1, 2, 1, 3, 1, 2, 4, 1, 1, 2, 3, 5, 1, 1, 2, 4, 1, 3, 1, 2, 1, 5, 1, 2, 3, 4, 6, 1, 1, 2, 1, 3, 1, 2, 4, 5, 1, 1, 2, 3, 6, 1, 1, 2, 4, 1, 3, 5, 1, 2, 1, 1, 2, 3

Offset: 1

Omar E. Pol, Jun 27 2009

If we define a divisor d|n to be inferior if d <= n/d, then inferior divisors are counted by A038548 and listed by this sequence. - Gus Wiseman, Mar 08 2021

			Triangle begins:
   1....... 1;
   2....... 1;
   3....... 1;
   4..... 1,2;
   5....... 1;
   6..... 1,2;
   7....... 1;
   8..... 1,2;
   9..... 1,3;
  10..... 1,2;
  11....... 1;
  12... 1,2,3;
  13....... 1;
  14..... 1,2;
  15..... 1,3;
  16... 1,2,4;

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened

Initial terms are A000012.
Final terms are A033676.
Row lengths are A038548 (number of inferior divisors).
Row sums are A066839 (sum of inferior divisors).
The prime terms are counted by A063962.
The odd terms are counted by A069288.
Row products are A072499.
Row LCMs are A072504.
The superior version is A161908.
The squarefree terms are counted by A333749.
The prime-power terms are counted by A333750.
The strictly superior version is A341673.
The strictly inferior version is A341674.
A001221 counts prime divisors, with sum A001414.
A000005 counts divisors, listed by A027750 with sum A000203.
A056924 count strictly superior (or strictly inferior divisors).
A207375 lists central divisors.
- Inferior: A217581.
- Superior: A033677, A051283, A059172, A063538, A063539, A070038, A116882, A116883, A341591, A341592, A341593, A341675, A341676.
- Strictly Inferior: A060775, A070039, A333805, A333806, A341596, A341677.
- Strictly Superior: A048098, A064052, A140271, A238535, A341594, A341595, A341642, A341643, A341644, A341645, A341646.
Cf. A000196, A001055, A001248, A006530, A020639, A050320, A068101, A161901.

a161906 n k = a161906_tabf !! (n-1) !! (k-1)
a161906_row n = a161906_tabf !! (n-1)
a161906_tabf = zipWith (\m ds -> takeWhile ((<= m) . (^ 2)) ds)
                       [1..] a027750_tabf'
-- Reinhard Zumkeller, Jun 24 2015, Mar 08 2013

div[n_] := Select[Divisors[n], # <= Sqrt[n] &]; div /@ Range[48] // Flatten (* Amiram Eldar, Nov 13 2020 *)

row(n) = select(x->(x<=sqrt(n)), divisors(n)); \\ Michel Marcus, Nov 13 2020

More terms from Sean A. Irvine, Nov 29 2010

A072499 Product of divisors of n which are <= n^(1/2).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 6, 1, 2, 3, 8, 1, 6, 1, 8, 3, 2, 1, 24, 5, 2, 3, 8, 1, 30, 1, 8, 3, 2, 5, 144, 1, 2, 3, 40, 1, 36, 1, 8, 15, 2, 1, 144, 7, 10, 3, 8, 1, 36, 5, 56, 3, 2, 1, 720, 1, 2, 21, 64, 5, 36, 1, 8, 3, 70, 1, 1152, 1, 2, 15, 8, 7, 36, 1, 320, 27, 2, 1, 1008, 5, 2, 3, 64, 1

Offset: 1

Amarnath Murthy, Jul 20 2002

nonn

a(1) = 1 and a(24) = 24. For each pair of primes p,q such that p < q < p^2, if n = p^3*q, then a(n) = n. There are others as well; e.g., a(40) = 40. - Don Reble, Aug 02 2002
Row products of the table in A161906. - Reinhard Zumkeller, Mar 08 2013
It appears that the fixed points belong to 3 categories: p^6 (A030516), p^3*q, or p*q*r. - Michel Marcus, May 16 2014

			a(20) = 8. The divisors of 20 are 1,2,4,5,10 and 20. a(20) = 1*2*4 = 8.

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

Cf. A072500, A072501, A161906.

Cf. A072504, A066839.

a072499 = product . a161906_row  -- Reinhard Zumkeller, Mar 08 2013

a[n_] := Times @@ Select[Divisors[n], #^2 <= n &]; Array[a, 100] (* Amiram Eldar, Jul 31 2022 *)

a(n) = my(d = divisors(n)); prod(i=1, #d, if (d[i]^2 <= n, d[i], 1)); \\ Michel Marcus, May 16 2014

from math import prod
from itertools import takewhile
from sympy import divisors
def A072499(n): return prod(takewhile(lambda x:x**2<=n,divisors(n))) # Chai Wah Wu, Dec 19 2023

More terms from Sascha Kurz, Feb 02 2003

A072505 a(n) = n / (LCM of divisors of n which are <= sqrt(n)).

Original entry on oeis.org

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

Offset: 1

Amarnath Murthy, Jul 20 2002

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

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

Cf. A072504.

f:= proc(n) n/ilcm(op(select(t -> t^2 <= n, numtheory:-divisors(n)))) end proc:
map(f, [$1..100]); # Robert Israel, Mar 19 2018

lc[n_]:=Module[{c=Select[Divisors[n],#<=Sqrt[n]&]},n/LCM@@c]; Array[lc,90] (* Harvey P. Dale, May 18 2012 *)

From Robert Israel, Mar 19 2018: (Start)
If n = p^k for prime p, then a(n) = p^ceiling(k/2).
In particular, a(n) = n if and only if n is prime.
If n = p*q for primes p < q, then a(n) = q. (End)

Corrected and extended by Matthew Conroy, Sep 09 2002

A372833 a(n) is the denominator of Sum_{d|n, d <= sqrt(n)} 1/d.

Original entry on oeis.org

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, 4, 1, 2, 3, 20, 1, 1, 1, 4, 15, 2, 1, 4, 7, 10, 3, 4, 1, 1, 5, 28, 3, 2, 1, 20, 1, 2, 21, 8, 5, 1, 1, 4, 3, 70, 1, 8, 1, 2, 15, 4, 7, 1, 1, 40

Offset: 1

Ilya Gutkovskiy, May 14 2024

a(n) is a divisor of A072504(n). The first few values of n for which a(n) != A072504(n) are 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 110, ... . - Pontus von Brömssen, May 15 2024

			1, 1, 1, 3/2, 1, 3/2, 1, 3/2, 4/3, 3/2, 1, 11/6, 1, 3/2, 4/3, 7/4, 1, 11/6, ...

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A017666, A066839, A072504, A372832 (numerators).

nmax = 80; CoefficientList[Series[Sum[x^(k^2)/(k (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest

a(n) = denominator(sumdiv(n, d, if (d^2 <= n, 1/d))); \\ Michel Marcus, May 14 2024

Denominators of coefficients in expansion of Sum_{k>=1} x^(k^2) / (k * (1 - x^k)).

cp's OEIS Frontend

A272533 Numbers n such that A072499(n)!=A072504(n).

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A161906 Triangle read by rows in which row n lists the divisors of n that are <= sqrt(n).

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A072499 Product of divisors of n which are <= n^(1/2).

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A072505 a(n) = n / (LCM of divisors of n which are <= sqrt(n)).

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A372833 a(n) is the denominator of Sum_{d|n, d <= sqrt(n)} 1/d.

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