A266531
Square array read by antidiagonals upwards: T(n,k) = n-th number with k odd divisors.
Original entry on oeis.org
1, 2, 3, 4, 5, 9, 8, 6, 18, 15, 16, 7, 25, 21, 81, 32, 10, 36, 27, 162, 45, 64, 11, 49, 30, 324, 63, 729, 128, 12, 50, 33, 625, 75, 1458, 105, 256, 13, 72, 35, 648, 90, 2916, 135, 225, 512, 14, 98, 39, 1250, 99, 5832, 165, 441, 405, 1024, 17, 100, 42, 1296, 117, 11664, 189, 450, 567, 59049, 2048, 19, 121, 51, 2401, 126, 15625
Offset: 1
The corner of the square array begins:
1, 3, 9, 15, 81, 45, 729, 105, 225, 405, ...
2, 5, 18, 21, 162, 63, 1458, 135, 441, 567, ...
4, 6, 25, 27, 324, 75, 2916, 165, 450, 810, ...
8, 7, 36, 30, 625, 90, 5832, 189, 882, 891, ...
16, 10, 49, 33, 648, 99, 11664, 195, 900, 1053, ...
32, 11, 50, 35, 1250, 117, 15625, 210, 1089, 1134, ...
64, 12, 72, 39, 1296, 126, 23328, 231, 1225, 1377, ...
128, 13, 98, 42, 2401, 147, 31250, 255, 1521, 1539, ...
...
Cf.
A001227,
A182469,
A236104,
A237591,
A237593,
A240062,
A261697,
A261698,
A261699,
A279387,
A286000,
A286001,
A296508.
A069289
Sum of odd divisors of n <= sqrt(n).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 6, 1, 4, 1, 1, 9, 1, 1, 4, 1, 6, 4, 1, 1, 4, 6, 1, 4, 1, 1, 9, 1, 1, 4, 8, 6, 4, 1, 1, 4, 6, 8, 4, 1, 1, 9, 1, 1, 11, 1, 6, 4, 1, 1, 4, 13, 1, 4, 1, 1, 9, 1, 8, 4, 1, 6, 13, 1, 1, 11, 6, 1, 4, 1
Offset: 1
-
a069289 n = sum $ takeWhile (<= a000196 n) $ a182469_row n
-- Reinhard Zumkeller, Apr 05 2015
-
Table[Total[Select[Divisors[n],OddQ[#]&&#<=Sqrt[n]&]],{n,120}] (* Harvey P. Dale, Jul 16 2017 *)
-
a(n) = my(ir = sqrtint(n)); sumdiv(n, d, (d % 2) * (d <= ir) * d); \\ Michel Marcus, Jan 14 2014
A136655
Product of odd divisors of n.
Original entry on oeis.org
1, 1, 3, 1, 5, 3, 7, 1, 27, 5, 11, 3, 13, 7, 225, 1, 17, 27, 19, 5, 441, 11, 23, 3, 125, 13, 729, 7, 29, 225, 31, 1, 1089, 17, 1225, 27, 37, 19, 1521, 5, 41, 441, 43, 11, 91125, 23, 47, 3, 343, 125, 2601, 13, 53, 729, 3025, 7, 3249, 29, 59, 225, 61, 31, 250047, 1, 4225, 1089
Offset: 1
-
a136655 = product . a182469_row -- Reinhard Zumkeller, May 01 2012
-
with(numtheory); f:=proc(n) local t1,i,k; t1:=divisors(n); k:=1; for i in t1 do if i mod 2 = 1 then k:=k*i; fi; od; k; end; # N. J. A. Sloane, Jul 14 2008
-
Array[Times @@ Select[Divisors@ #, OddQ] &, 66] (* Michael De Vlieger, Aug 03 2017 *)
a[n_] := (oddpart = n/2^IntegerExponent[n, 2])^(DivisorSigma[0, oddpart]/2); Array[a, 100] (* Amiram Eldar, Jun 26 2022 *)
-
a(n) = my(d=divisors(n)); prod(k=1, #d, if (d[k]%2, d[k], 1)); \\ Michel Marcus, Aug 04 2017
-
from math import isqrt
from sympy import divisor_count
def A136655(n):
d = divisor_count(m:=n>>(~n&n-1).bit_length())
return isqrt(m)**d if d&1 else m**(d>>1) # Chai Wah Wu, Jun 27 2025
A261697
Irregular triangle read by rows in which row n lists the odd divisors of n in the ordering given by A261699.
Original entry on oeis.org
1, 1, 1, 3, 1, 1, 5, 1, 3, 1, 7, 1, 1, 9, 3, 1, 5, 1, 11, 1, 3, 1, 13, 1, 7, 1, 15, 3, 5, 1, 1, 17, 1, 3, 9, 1, 19, 1, 5, 1, 21, 3, 7, 1, 11, 1, 23, 1, 3, 1, 25, 5, 1, 13, 1, 27, 3, 9, 1, 7, 1, 29, 1, 3, 15, 5, 1, 31, 1, 1, 33, 3, 11, 1, 17, 1, 35, 5, 7, 1, 3, 9, 1, 37, 1, 19, 1, 39, 3, 13, 1, 5, 1, 41, 1, 3, 21, 7, 1, 43, 1, 11, 1, 45, 3, 5, 15, 9
Offset: 1
List of divisors of 45 from distinct sequences:
45th row of triangle A182469: 1, 3, 5, 9, 15, 45.
45th row of triangle A261698: 1, 45, 3, 15, 5, 9.
45th row of this triangle...: 1, 45, 3, 5, 15, 9.
Triangle begins:
1;
1;
1, 3;
1;
1, 5;
1, 3;
1, 7;
1;
1, 9, 3;
1, 5;
1, 11;
1, 3;
1, 13;
1, 7;
1, 15, 3, 5;
1;
1, 17;
1, 3, 9;
1, 19;
1, 5;
1, 21, 3, 7;
...
A379379
Sum of odd divisors of n except the "e" odd divisors described in A005279.
Original entry on oeis.org
1, 1, 4, 1, 6, 1, 8, 1, 13, 6, 12, 1, 14, 8, 19, 1, 18, 1, 20, 1, 32, 12, 24, 1, 31, 14, 40, 1, 30, 1, 32, 1, 48, 18, 41, 1, 38, 20, 56, 1, 42, 1, 44, 12, 49, 24, 48, 1, 57, 31, 72, 14, 54, 1, 72, 1, 80, 30, 60, 1, 62, 32, 95, 1, 84, 1, 68, 18, 96, 41, 72, 1, 74
Offset: 1
Cf.
A001227,
A000593,
A005279,
A182469,
A237270,
A237271,
A237593,
A239657,
A379374,
A379384,
A379461.
-
a[n_] := Module[{d = Partition[Divisors[n], 2, 1]}, 1 + Total[Select[d, OddQ[#[[2]]] && #[[2]] >= 2*#[[1]] &][[;; , 2]]]]; Array[a, 100] (* Amiram Eldar, Dec 22 2024 *)
A037283
Replace n with concatenation of its odd divisors.
Original entry on oeis.org
1, 1, 13, 1, 15, 13, 17, 1, 139, 15, 111, 13, 113, 17, 13515, 1, 117, 139, 119, 15, 13721, 111, 123, 13, 1525, 113, 13927, 17, 129, 13515, 131, 1, 131133, 117, 15735, 139, 137, 119, 131339, 15, 141, 13721, 143, 111, 13591545, 123, 147, 13, 1749, 1525, 131751
Offset: 1
-
a037283 = read . concat . (map show) . a182469_row :: Integer -> Integer
-- Reinhard Zumkeller, May 01 2012
-
dtn[ L_ ] := Fold[ 10#1+#2&, 0, L ] Array[ dtn[ Flatten[ Map[ IntegerDigits, Select[ Divisors[ # ], OddQ ] ] ] ]&, 50 ]
cod[n_]:=FromDigits[Flatten[IntegerDigits/@Select[Divisors[n],OddQ]]]; Array[cod,60] (* Harvey P. Dale, Jan 24 2014 *)
-
from sympy import divisors
def a(n): return int("".join(str(d) for d in divisors(n) if d%2==1))
print([a(n) for n in range(1, 52)]) # Michael S. Branicky, Dec 31 2020
A037284
Replace n with concatenation of its odd divisors >1.
Original entry on oeis.org
0, 0, 3, 0, 5, 3, 7, 0, 39, 5, 11, 3, 13, 7, 3515, 0, 17, 39, 19, 5, 3721, 11, 23, 3, 525, 13, 3927, 7, 29, 3515, 31, 0, 31133, 17, 5735, 39, 37, 19, 31339, 5, 41, 3721, 43, 11, 3591545, 23, 47, 3, 749, 525, 31751
Offset: 1
-
a037284 n
| a209229 n == 1 = 0
| otherwise = read $ concat $ (map show) $ tail $ a182469_row n
-- Reinhard Zumkeller, May 01 2012
-
Array[FromDigits[Flatten[IntegerDigits/@Rest[Select[Divisors[#], OddQ]]]]&, 60] (* Harvey P. Dale, Mar 03 2014 *)
-
from sympy import divisors
def a(n):
odd_divisors_gt1 = [d for d in divisors(n)[1:] if d%2 == 1]
if len(odd_divisors_gt1) == 0: return 0
else: return int("".join(str(d) for d in odd_divisors_gt1))
print([a(n) for n in range(1, 52)]) # Michael S. Branicky, Dec 31 2020
A037285
Replace n with concatenation of its nontrivial odd divisors.
Original entry on oeis.org
0, 0, 0, 0, 0, 3, 0, 0, 3, 5, 0, 3, 0, 7, 35, 0, 0, 39, 0, 5, 37, 11, 0, 3, 5, 13, 39, 7, 0, 3515, 0, 0, 311, 17, 57, 39, 0, 19, 313, 5, 0, 3721, 0, 11, 35915, 23, 0, 3, 7, 525, 317, 13, 0, 3927, 511, 7, 319, 29, 0, 3515, 0, 31, 37921, 0, 513, 31133, 0, 17, 323, 5735, 0, 39, 0
Offset: 1
-
import Data.List (delete)
a037285 n
| a209229 n == 1 = 0
| a010051 n == 1 = 0
| otherwise = read $ concat $ (map show) $ delete n $ tail $ a182469_row n
-- Reinhard Zumkeller, May 01 2012
-
from sympy import divisors
def a(n):
nontrivial_odd_divisors = [d for d in divisors(n)[1:-1] if d%2 == 1]
if len(nontrivial_odd_divisors) == 0: return 0
else: return int("".join(str(d) for d in nontrivial_odd_divisors))
print([a(n) for n in range(1, 70)]) # Michael S. Branicky, Dec 31 2020
A053636
a(n) = Sum_{odd d|n} phi(d)*2^(n/d).
Original entry on oeis.org
0, 2, 4, 12, 16, 40, 72, 140, 256, 540, 1040, 2068, 4128, 8216, 16408, 32880, 65536, 131104, 262296, 524324, 1048640, 2097480, 4194344, 8388652, 16777728, 33554600, 67108912, 134218836, 268435552, 536870968, 1073744160, 2147483708
Offset: 0
2*x + 4*x^2 + 12*x^3 + 16*x^4 + 40*x^5 + 72*x^6 + 140*x^7 + 256*x^8 + 540*x^9 + ...
-
a053636 0 = 0
a053636 n = sum $ zipWith (*) (map a000010 ods) (map ((2 ^) . (div n)) ods)
where ods = a182469_row n
-- Reinhard Zumkeller, Sep 13 2013
-
a[ n_] := If[ n < 1, 0, Sum[ Mod[ d, 2] EulerPhi[ d] 2^(n / d), {d, Divisors[ n]}]] (* Michael Somos, May 09 2013 *)
-
{a(n) = if( n<1, 0, sumdiv( n, d, (d % 2) * eulerphi(d) * 2^(n / d)))} /* Michael Somos, May 09 2013 */
-
from sympy import totient, divisors
def A053636(n): return (sum(totient(d)<>(~n&n-1).bit_length(),generator=True))<<1) # Chai Wah Wu, Feb 21 2023
A379630
Irregular triangle read by rows in which row n lists the smallest parts of the partitions of n into consecutive parts followed by the conjugate corresponding odd divisors of n in accordance with the theorem of correspondence described in the Comments lines.
Original entry on oeis.org
1, 1, 2, 1, 3, 1, 3, 1, 4, 1, 5, 2, 5, 1, 6, 1, 3, 1, 7, 3, 7, 1, 8, 1, 9, 4, 2, 3, 9, 1, 10, 1, 5, 1, 11, 5, 11, 1, 12, 3, 3, 1, 13, 6, 13, 1, 14, 2, 7, 1, 15, 7, 4, 1, 5, 3, 15, 1, 16, 1, 17, 8, 17, 1, 18, 5, 3, 9, 3, 1, 19, 9, 19, 1, 20, 2, 5, 1, 21, 10, 6, 1, 7, 3, 21, 1, 22, 4, 11, 1, 23, 11, 23, 1, 24, 7, 3, 1
Offset: 1
Triangle begins:
1, 1;
2, 1;
3, 1, 3, 1;
4, 1;
5, 2, 5, 1;
6, 1, 3, 1;
7, 3, 7, 1;
8, 1;
9, 4, 2, 3, 9, 1;
10, 1, 5, 1;
11, 5, 11, 1;
12, 3, 3, 1;
13, 6, 13, 1;
14, 2, 7, 1;
15, 7, 4, 1, 5, 3, 15, 1;
16, 1;
17, 8, 17, 1;
18, 5, 3, 9, 3, 1;
19, 9, 19, 1;
20, 2, 5, 1;
21, 10, 6, 1, 7, 3, 21, 1;
...
For n = 21 the partitions of 21 into consecutive parts are [21], [11, 10], [8, 7, 6], [6, 5, 4, 3, 2, 1].
On the other hand the odd divisors of 21 are [1, 3, 7, 21].
To determine how these partitions are related to the odd divisors we follow the two rules of the theorem as shown below:
The first partition is [21] and the number of parts is 1 and 1 is odd so the corresponding odd divisor of 21 is 1.
The second partition is [11, 10] and the number of parts is 2 and 2 even so the corresponding odd divisor of 21 is equal to 11 + 10 = 21.
The third partition is [8, 7, 6] and the number of parts is 3 and 3 is odd so the corresponding odd divisor of 21 is 3.
The fourth partition is [6, 5, 4, 3, 2, 1] and the number of parts is 6 and 6 is even so the corresponding odd divisor of 21 is equal to 6 + 1 = 5 + 2 = 4 + 3 = 7.
Summarizing in a table:
--------------------------------------
Correspondence
--------------------------------------
Partitions of 21 Odd
into consecutive divisors
parts of 21
------------------- ----------
[21] .................... 1
[11, 10] ................ 21
[8, 7, 6] ................ 3
[6, 5, 4, 3, 2, 1] ....... 7
.
Then we can make a table of conjugate correspondence in which the four partitions are arrenged in four columns with the smallest parts at the top as shown below:
------------------------------------------
Conjugate correspondence
------------------------------------------
Partitions of 21 Odd
into consecutive divisors
parts as columns of 21
------------------- ------------------
21 10 6 1 7 3 21 1
| 11 7 2 | | | |
| | 8 3 | | | |
| | | 4 | | | |
| | | 5 | | | |
| | | 6 | | | |
| | | |_______| | | |
| | |_________________| | |
| |___________________________| |
|_____________________________________|
.
Then removing all rows except the first row we have a table of conjugate correspondence for smallest parts and odd divisors as shown below:
------------------- ------------------
Smallest parts Odd divisors
------------------- ------------------
21 10 6 1 7 3 21 1
| | | |_______| | | |
| | |_________________| | |
| |___________________________| |
|_____________________________________|
.
So the 21st row of the triangle is [21, 10, 6, 1, 7, 3, 21, 1].
.
Illustration of initial terms in an isosceles triangle demonstrating the theorem:
. _ _
_|1|1|_
_|2 _|_ 1|_
_|3 |1|3| 1|_
_|4 _| | |_ 1|_
_|5 |2 _|_ 5| 1|_
_|6 _| |1|3| |_ 1|_
_|7 |3 | | | 7| 1|_
_|8 _| _| | |_ |_ 1|_
_|9 |4 |2 _|_ 3| 9| 1|_
_|10 _| | |1|5| | |_ 1|_
_|11 |5 _| | | | |_ 11| 1|_
_|12 _| |3 | | | 3| |_ 1|_
_|13 |6 | _| | |_ | 13| 1|_
_|14 _| _| |2 _|_ 7| |_ |_ 1|_
_|15 |7 |4 | |1|5| | 3| 15| 1|_
_|16 _| | | | | | | | |_ 1|_
_|17 |8 _| _| | | | |_ |_ 17| 1|_
_|18 _| |5 |3 | | | 9| 3| |_ 1|_
_|19 |9 | | _| | |_ | | 19| 1|_
_|20 _| _| | |2 _|_ 5| | |_ |_ 1|_
|21 |10 |6 | | |1|7| | | 3| 21| 1|
.
The geometrical structure of the above isosceles triangle is defined in A237593. See also the triangle A286001.
Note that the diagram also can be interpreted as a template which after folding gives a 90 degree pop-up card which has essentially the same structure as the stepped pyramid described in A245092.
.
Cf.
A196020,
A204217,
A235791,
A236104,
A237048,
A237270,
A237271,
A237591,
A237593,
A245092,
A262626,
A341971.
Comments