A079277 -id:A079277 - cp's OEIS frontend

A285708 a(n) = n / A285707(n).

1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 4, 13, 7, 5, 2, 17, 9, 19, 5, 7, 11, 23, 4, 5, 13, 3, 7, 29, 10, 31, 2, 11, 17, 7, 9, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 4, 7, 5, 17, 13, 53, 9, 11, 8, 19, 29, 59, 10, 61, 31, 9, 2, 13, 33, 67, 17, 23, 35, 71, 9, 73, 37, 5, 19, 11, 13, 79, 5, 3, 41, 83, 28, 17, 43, 29, 11, 89, 10, 13, 23, 31, 47, 19, 32, 97, 49, 11, 5

Offset: 1

Antti Karttunen, Apr 26 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A079277, A285707.

from sympy import divisors, gcd
from sympy.ntheory.factor_ import core
def a007947(n): return max(i for i in divisors(n) if core(i) == i)
def a079277(n):
    k=n - 1
    while True:
        if a007947(k*n) == a007947(n): return k
        else: k-=1
def a285707(n): return 1 if n==1 else gcd(n, a079277(n))
print([n//a285707(n) for n in range(1, 151)]) # Indranil Ghosh, Apr 27 2017

Scheme
```
(define (A285708 n) (/ n (A285707 n)))
```

a(n) = n / A285707(n).

For n > 1, a(n) = n / gcd(n, A079277(n)).

A335717 a(n) is the smallest dividend m of the Euclidean division m = d*n + r such that m/d = r/n.

Original entry on oeis.org

25, 100, 162, 676, 400, 2500, 1156, 2352, 1458, 14884, 2601, 28900, 5202, 6084, 8712, 84100, 9408, 131044, 10816, 22500, 22275, 280900, 19602, 79380, 36517, 60516, 40000, 708964, 31827, 925444, 67600, 46128, 80937, 62500, 55112, 1876900, 112651, 88752, 83232

Offset: 2

Stéphane Rézel, Jun 18 2020

Such a Euclidean division exists, for all n>1, with r = n^2 + 1 because then m = r^2 and d = n*r. This solution leads to the largest possible dividend, given (for n>1) by A082044(n). This is also the only solution when n is prime, then a(n) = A082044(n).

Solutions with r = n^2 + k are only possible for most of the integers k < n such that any prime factor of k is also a prime factor of n. The largest such integers k are given by A079277(n). In the present sequence, k is not always this largest integer: a(18) is defined by k = 12 while it cannot be defined by k = 16 = A079277(18).

			a(2) = 25 because 25 = 10*2 + 5 and 25/10 = 5/2. Furthermore, there is no Euclidean division with a dividend less than 25 such that m/d = r/2.
a(4) = 162 because 162 = 36*4 + 18 and 162/36 = 18/4. The other Euclidean division with the same property has a larger dividend : 289 = 68*4 + 17 and 289/68 = 17/4.

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

Cf. A079277, A082044.

f:= proc(n) local m, r,k,d;
  for k from n-1 to 1 by -1 do
    r:= n^2+k;
    m:= r^2/k;
    if not m::integer then next fi;
    d:= n*r/k;
    if d::integer then return m fi;
  od
end proc:
map(f, [$2..50]); # Robert Israel, Sep 16 2020

f[n_] := Module[{m, r, k, d}, For[k = n-1, k >= 1, k--, r = n^2+k; m = r^2/k; If[!IntegerQ[m], Continue]; d = n*r/k; If[IntegerQ[d], Return[m]]]];
Table[f[n], {n, 2, 50}] (* Jean-François Alcover, Jul 31 2024, after Robert Israel *)

a(n) = k=n; until(k==1, k--; r=k + n^2; d=n*r/k; m=r^2/k; if(floor(d)==d && floor(m)==m, return(m); break));

cp's OEIS Frontend

A285708 a(n) = n / A285707(n).

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A335717 a(n) is the smallest dividend m of the Euclidean division m = d*n + r such that m/d = r/n.

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