A097448 -id:A097448 - cp's OEIS frontend

A207375 Irregular array read by rows in which row n lists the (one or two) central divisors of n in increasing order.

1, 1, 2, 1, 3, 2, 1, 5, 2, 3, 1, 7, 2, 4, 3, 2, 5, 1, 11, 3, 4, 1, 13, 2, 7, 3, 5, 4, 1, 17, 3, 6, 1, 19, 4, 5, 3, 7, 2, 11, 1, 23, 4, 6, 5, 2, 13, 3, 9, 4, 7, 1, 29, 5, 6, 1, 31, 4, 8, 3, 11, 2, 17, 5, 7, 6, 1, 37, 2, 19, 3, 13, 5, 8, 1, 41, 6, 7, 1, 43

Offset: 1

Omar E. Pol, Feb 23 2012

If n is a square then row n lists only the square root of n because the squares (A000290) have only one central divisor.
If n is not a square then row n lists the pair (j, k) of divisors of n, nearest to the square root of n, such that j*k = n.
Conjecture 1: It appears that the n-th record in this sequence is the last member of row A008578(n).
Column 1 gives A033676. Right border gives A033677. - Omar E. Pol, Feb 26 2019
The conjecture 1 follows from Bertrand's Postulate. - Charles R Greathouse IV, Feb 11 2022
Row products give A097448. - Omar E. Pol, Feb 17 2022

			Array begins:
  1;
  1,  2;
  1,  3;
  2;
  1,  5;
  2,  3;
  1,  7;
  2,  4;
  3;
  2,  5;
  1, 11;
  3,  4;
  1, 13;
...

Alois P. Heinz, Rows n = 1..5000
Omar E. Pol, Illustration of the divisors of the first 12 positive integers

Row n has length A169695(n).
Row sums give A207376.
Cf. A000005, A000290, A008578, A027750, A033676, A033677, A097448, A161901, A161904.

A207375row[n_] := ArrayPad[#, -Floor[(Length[#] - 1)/2]] & [Divisors[n]];
Array[A207375row, 50] (* Paolo Xausa, Apr 07 2025 *)

A088835 a(n) = lcm(A020639(n), A006530(n)).

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 10, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 41, 14, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 10, 61, 62, 21, 2, 65, 22, 67, 34, 69, 14, 71, 6, 73, 74, 15, 38, 77

Offset: 1

Labos Elemer, Oct 31 2003

nonn

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000

Cf. A020639, A006530, A066048, A007947, A062953, A015052, A015053, A053166, A097448.

Map[LCM[#[[1,1]], #[[-1,1]]] &, FactorInteger[Range[100]]] (* Paolo Xausa, Mar 29 2024 *)

a(n) = if (n==1, 1, my(f=factor(n)[,1]~); lcm(vecmin(f), vecmax(f))); \\ Michel Marcus, Mar 15 2018

a(n) = A097448(A066048(n)). - Amiram Eldar, Mar 16 2025

A097449 If n is a cube, replace it with the cube root of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 2, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 3, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 4, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 0

Cino Hilliard, Aug 23 2004

			The 9th integer is 8 so a(9) = 8^(1/3) = 2.

Cf. A016627, A097448, A197070.

rcr[n_]:=Module[{crn=Power[n, (3)^-1]},If[IntegerQ[crn],crn,n]]; Array[ rcr,80,0] (* Harvey P. Dale, Jan 28 2012 *)

iscube(n) = { local(r); r = n^(1/3); if(floor(r+.5)^3== n,1,0) }
replcube(n) = { for(x=0,n, if(iscube(x),y=x^(1/3),y=x); print1(floor(y)",")) }

a(n)=ispower(n,3,&n);n \\ Charles R Greathouse IV, Oct 27 2011

Sum_{n>=1} (-1)^(n+1)/n = 2*log(2) - 3*zeta(3)/4 = A016627 - A197070. - Amiram Eldar, Jul 07 2024

Corrected by T. D. Noe, Oct 25 2006

A361253 If n = m^2 for some m > 1 then a(n) = a(m), otherwise a(n) = n.

Original entry on oeis.org

0, 1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 12, 13, 14, 15, 2, 17, 18, 19, 20, 21, 22, 23, 24, 5, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 7, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 8, 65, 66, 67, 68

Offset: 0

Rémy Sigrist, Mar 06 2023

All terms belong to A000037 U { 0, 1 }.
All terms of A000037 appear infinitely many times.
This sequence can be seen as the limit of the k-th iterate of A097448 as k tends to infinity.

			a(9) = a(3^2) = a(3) = 3 (as 3 is not a square).

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A000037, A001146, A011764, A176594, A097448.

nn = 120; Array[Set[a[#], #] &, 2, 0]; Do[If[IntegerQ[#], Set[k, a[#]], Set[k, n]] &[Sqrt[n]]; Set[a[n], k], {n, nn}]; Array[a, nn] (* Michael De Vlieger, Mar 06 2023 *)

a(n) = my (m); { while (n > 1 && issquare(n, &m), n = m); return (n) }

from sympy import integer_nthroot
def A361253(n):
    if n <= 1:
        return n
    a, b = integer_nthroot(c:=n,2)
    while b:
        a, b = integer_nthroot(c:=a,2)
    return c # Chai Wah Wu, Mar 17 2023

a(a(n)) = a(n).
a(n) <= A097448(n).
a(n) = 2 iff n belongs to A001146.
a(n) = 3 iff n belongs to A011764.
a(n) = 5 iff n belongs to A176594.

A216455 If n is a perfect fourth power, then replace it with the fourth root of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 2, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 3, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

Offset: 1

V. Raman, Sep 07 2012

Cf. A097448, A097449.

a(n)=ispower(n,4,&n); n \\ Charles R Greathouse IV, Sep 01 2015

cp's OEIS Frontend

A207375 Irregular array read by rows in which row n lists the (one or two) central divisors of n in increasing order.

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Mathematica

A088835 a(n) = lcm(A020639(n), A006530(n)).

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A097449 If n is a cube, replace it with the cube root of n.

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A361253 If n = m^2 for some m > 1 then a(n) = a(m), otherwise a(n) = n.

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A216455 If n is a perfect fourth power, then replace it with the fourth root of n.

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