A093803 -id:A093803 - cp's OEIS frontend

A358033 a(1) = 2; a(n) - a(n-1) = A093803(a(n-1)), the largest odd proper divisor of a(n-1).

2, 3, 4, 5, 6, 9, 12, 15, 20, 25, 30, 45, 60, 75, 100, 125, 150, 225, 300, 375, 500, 625, 750, 1125, 1500, 1875, 2500, 3125, 3750, 5625, 7500, 9375, 12500, 15625, 18750, 28125, 37500, 46875, 62500, 78125, 93750, 140625, 187500, 234375, 312500, 390625, 468750

Offset: 1

Eric Angelini and Gavin Lupo, Oct 25 2022

			a(1) = 2.
a(2) = 3. The only proper divisor of 2 is 1; 2 + 1 = 3.
a(3) = 4. The only proper divisor of 3 is 1; 3 + 1 = 4.
...
a(8) = 15.
a(9) = 20. Proper divisors of 15 are 1, 3, 5; largest odd proper divisor = 5; 15 + 5 = 20.

Cf. A093803, A000792 (with largest proper divisor instead).

Cf. A027750, A038754, A056487, A356639.

f(n) = my(x=if(n==1, 1, n/factor(n)[1, 1])); x >> valuation(x, 2); \\ Michel Marcus, Oct 26 2022
lista(nn) = my(va = vector(nn)); va[1] = 2; for (n=2, nn, va[n] = va[n-1] + f(va[n-1]);); va; \\ Michel Marcus, Oct 26 2022

a_n = 2
result = [2]
for n in range(30):
    temp = []
    for i in range(1, a_n):
        if a_n % i == 0:
            if (i % 2 != 0) and (i != a_n):
                temp.append(i)
    result.append(a_n + max(temp))
    a_n = a_n + max(temp)
print(result)

a(n+1) - a(n) = A056487(floor((n-2)/3)), for n > 2. This works because A056487(n+3) = A056487(n+2)*A056487(n+1)/A056487(n). - Thomas Scheuerle, Oct 26 2022

A366519 Largest odd divisor of n which is <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 5, 1, 3, 1, 1, 5, 1, 1, 3, 1, 5, 3, 1, 1, 3, 5, 1, 3, 1, 1, 5, 1, 1, 3, 7, 5, 3, 1, 1, 3, 5, 7, 3, 1, 1, 5, 1, 1, 7, 1, 5, 3, 1, 1, 3, 7, 1, 3, 1, 1, 5, 1, 7, 3, 1, 5, 9, 1, 1, 7, 5, 1, 3, 1, 1, 9, 7, 1, 3, 1, 5, 3, 1, 7, 9, 5

Offset: 1

Ilya Gutkovskiy, Oct 11 2023

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A000265, A033676, A093803, A366520.

Table[Last[Select[Divisors[n], # <= Sqrt[n] && OddQ[#] &]], {n, 100}]

a(n) = {my(m=1); fordiv(n/2^valuation(n,2), d, if(d^2 <= n, m=max(m,d))); m} \\ Andrew Howroyd, Oct 11 2023

A366520 Largest odd divisor of n which is < sqrt(n), for n >= 2; a(1) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 5, 1, 1, 3, 1, 5, 3, 1, 1, 3, 5, 1, 3, 1, 1, 5, 1, 1, 3, 1, 5, 3, 1, 1, 3, 5, 7, 3, 1, 1, 5, 1, 1, 7, 1, 5, 3, 1, 1, 3, 7, 1, 3, 1, 1, 5, 1, 7, 3, 1, 5, 3, 1, 1, 7, 5, 1, 3, 1, 1, 9, 7, 1, 3, 1, 5, 3, 1, 7, 9, 5

Offset: 1

Ilya Gutkovskiy, Oct 11 2023

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A000265, A060775, A093803, A366519.

Join[{1}, Table[Last[Select[Divisors[n], # < Sqrt[n] && OddQ[#] &]], {n, 2, 100}]]

a(n) = {my(m=1); fordiv(n/2^valuation(n,2), d, if(d^2 < n, m=max(m,d))); m} \\ Andrew Howroyd, Oct 11 2023

A254522 Numerators of (2^n - 1 + (-1)^n)/(2*n), n > 0.

Original entry on oeis.org

0, 1, 1, 2, 3, 16, 9, 16, 85, 256, 93, 512, 315, 4096, 5461, 2048, 3855, 65536, 13797, 131072, 349525, 1048576, 182361, 1048576, 3355443, 16777216, 22369621, 33554432, 9256395, 268435456, 34636833, 67108864, 1431655765, 4294967296, 17179869183, 8589934592, 1857283155, 68719476736, 91625968981

Offset: 1

Paul Curtz, Jan 31 2015

nonn

An autosequence of the first kind is a sequence which main diagonal is A000004.
Difference table of a(n)/A093803(n):
     0,     1,    1,   2,    3, 16/3, ...
     1,     0,    1,   1,  7/3, 11/3, ...
    -1,     1,    0, 4/3,  4/3, 10/3, ...
     2,    -1,  4/3,   0,    2,    2, ...
    -3,   7/3, -4/3,   2,    0, 16/5, ...
  16/3, -11/3, 10/3,  -2, 16/5,    0, ...
etc.
This is an autosequence of the first kind.
Its first (or second) upper diagonal is A075101(n)/(2*A000265(n)).
From Robert Israel, Apr 03 2017: (Start)
If p is a prime == 5 (mod 8), then a(5*p) = (2^(5*p-1)-1)/5 and a(5*p+3) = 2^(5*p) = 10*a(5*p)+2. This explains pairs such as
   a(25) = 3355443
   a(28) = 33554432
and
   a(65) = 3689348814741910323
   a(68) = 36893488147419103232. (End)

Robert Israel, Table of n, a(n) for n = 1..3321

Cf. A000004, A075101, A093803, A099430.

seq(numer((2^n-1+(-1)^n)/(2*n)), n=1..50); # Robert Israel, Feb 01 2015

Table[Numerator[(2^n - 1 + (-1)^n)/(2*n)], {n, 39}] (* Michael De Vlieger, Feb 01 2015 *)

a(25) corrected by Robert Israel, Apr 03 2017

A365837 Largest proper square divisor of n, for n >= 2; a(1) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 4, 1, 1, 1, 16, 1, 1, 1, 9, 1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 16, 1, 25, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 16, 1, 1, 1, 4, 1, 1, 1, 36, 1, 1, 25, 4, 1, 1, 1, 16, 9, 1, 1, 4, 1, 1, 1, 4, 1, 9, 1, 4, 1, 1, 1, 16, 1, 49, 9, 25

Offset: 1

Ilya Gutkovskiy, Oct 17 2023

nonn

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

Cf. A008833, A032742, A063928, A085392, A089384, A093803, A366524.

f:= proc(n) local F, t;
  if issqr(n) then
    n/min(numtheory:-factorset(n))^2
  else
    F:= ifactors(n)[2];
    mul(t[1]^(2*floor(t[2]/2)),t=F)
  fi
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Nov 20 2023

Join[{1}, Table[Last[Select[Divisors[n], # < n && IntegerQ[Sqrt[#]]  &]], {n, 2, 100}]]
f[p_, e_] := p^(2*Floor[e/2]); a[n_] := Module[{fct = FactorInteger[n]}, Times @@ f @@@ fct/If[AllTrue[fct[[;; , 2]], EvenQ], fct[[1, 1]]^2, 1]]; Array[a, 100] (* Amiram Eldar, Oct 19 2023 *)

a(n) = if (n==1, 1, my(d=divisors(n)); vecmax(select(issquare, Vec(d, #d-1)))); \\ Michel Marcus, Oct 17 2023

from math import prod
from sympy import factorint
def A365837(n):
    if n<=1: return 1
    f = factorint(n)
    return prod(p**(e&-2) for p, e in f.items())//(min(f)**2 if all(e&1^1 for e in f.values()) else 1) # Chai Wah Wu, Oct 20 2023

A366649 Largest prime power (including 1) proper divisor of n, for n >= 2; a(1) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 4, 1, 7, 5, 8, 1, 9, 1, 5, 7, 11, 1, 8, 5, 13, 9, 7, 1, 5, 1, 16, 11, 17, 7, 9, 1, 19, 13, 8, 1, 7, 1, 11, 9, 23, 1, 16, 7, 25, 17, 13, 1, 27, 11, 8, 19, 29, 1, 5, 1, 31, 9, 32, 13, 11, 1, 17, 23, 7, 1, 9, 1, 37, 25, 19, 11, 13, 1, 16, 27, 41, 1, 7, 17

Offset: 1

Ilya Gutkovskiy, Oct 17 2023

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

Cf. A032742, A034699, A063928, A085392, A089384, A093803.

f:= proc(n) local F,t;
  F:= ifactors(n)[2];
  if nops(F) = 1 then n/F[1,1]
  else max(map(t -> t[1]^t[2], F))
  fi
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Nov 19 2023

Join[{1}, Table[Last[Select[Divisors[n], # < n && (# == 1 || PrimePowerQ[#]) &]], {n, 2, 85}]]
a[n_] := Module[{f = FactorInteger[n]}, If[Length[f] == 1, f[[1, 1]]^(f[[1, 2]] - 1), Max[Power @@@ f]]]; Array[a, 100] (* Amiram Eldar, Oct 19 2023 *)

a(n) = if (n==1, 1, my(d=divisors(n)); vecmax(select(x->(isprimepower(x) || (x==1)), Vec(d, #d-1)))); \\ Michel Marcus, Oct 17 2023

cp's OEIS Frontend

A358033 a(1) = 2; a(n) - a(n-1) = A093803(a(n-1)), the largest odd proper divisor of a(n-1).

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A366519 Largest odd divisor of n which is <= sqrt(n).

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A366520 Largest odd divisor of n which is < sqrt(n), for n >= 2; a(1) = 1.

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A254522 Numerators of (2^n - 1 + (-1)^n)/(2*n), n > 0.

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A365837 Largest proper square divisor of n, for n >= 2; a(1) = 1.

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Maple

Mathematica

PARI

Python

A366649 Largest prime power (including 1) proper divisor of n, for n >= 2; a(1) = 1.

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Maple

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