A175317 -id:A175317 - cp's OEIS frontend

A280114 Partial sums of A175317 (Sum_{d|n} pod(d)).

1, 4, 8, 19, 25, 67, 75, 150, 181, 289, 301, 2079, 2093, 2299, 2533, 3632, 3650, 9551, 9571, 17687, 18139, 18637, 18661, 352279, 352410, 353102, 353862, 376028, 376058, 1186430, 1186462, 1220329, 1221433, 1222609, 1223847, 11309180, 11309218, 11310684

Offset: 1

Jaroslav Krizek, Dec 26 2016

nonn

pod(n) is the product of the divisors of n (A007955).

Cf. A007955, A175317, A206032, A280085 (partial sums of A206032), A280115 (partial products of A175317).

[&+[&+[&*[b: b in Divisors(d)]: d in Divisors(k)]: k in [1..n]]: n in [1..1000]];

Accumulate@ Array[Sum[Times @@ Divisors@ d, {d, Divisors@ #}] &, 38] (* Michael De Vlieger, Dec 26 2016 *)

a(n) = Sum_{i=1..n} A175317(i).

A280115 Partial products of A175317 (Sum_{d|n} pod(d)).

Original entry on oeis.org

1, 3, 12, 132, 792, 33264, 266112, 19958400, 618710400, 66820723200, 801848678400, 1425686950195200, 19959617302732800, 4111681164362956800, 962133392460931891200, 1057384598314564148428800, 19032922769662154671718400, 112313277263776374717810278400

Offset: 1

Jaroslav Krizek, Dec 26 2016

nonn

pod(n) is the product of the divisors of n (A007955).

Cf. A007955, A175317, A206032, A280086 (partial products of A206032), A280114 (partial sums of A175317).

[&*[&+[&*[b: b in Divisors(d)]: d in Divisors(k)]: k in [1..n]]: n in [1..1000]]

FoldList[Times[#1, #2] &, Array[Sum[Times @@ Divisors@ d, {d, Divisors@ #}] &, 18]] (* Michael De Vlieger, Dec 26 2016 *)

a(n) = Product_{i=1..n} A175317(i).

A028982 Squares and twice squares.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 18, 25, 32, 36, 49, 50, 64, 72, 81, 98, 100, 121, 128, 144, 162, 169, 196, 200, 225, 242, 256, 288, 289, 324, 338, 361, 392, 400, 441, 450, 484, 512, 529, 576, 578, 625, 648, 676, 722, 729, 784, 800, 841, 882, 900, 961, 968, 1024

Offset: 1

Patrick De Geest

Numbers n such that sum of divisors of n (A000203) is odd.
Also the numbers with an odd number of run sums (trapezoidal arrangements, number of ways of being written as the difference of two triangular numbers). - Ron Knott, Jan 27 2003
Pell(n)*Sum_{k|n} 1/Pell(k) is odd, where Pell(n) is A000129(n). - Paul Barry, Oct 12 2005
Number of odd divisors of n (A001227) is odd. - Vladeta Jovovic, Aug 28 2007
A071324(a(n)) is odd. - Reinhard Zumkeller, Jul 03 2008
Sigma(a(n)) = A000203(a(n)) = A152677(n). - Jaroslav Krizek, Oct 06 2009
Numbers n such that sum of odd divisors of n (A000593) is odd. - Omar E. Pol, Jul 05 2016
A187793(a(n)) is odd. - Timothy L. Tiffin, Jul 18 2016
If k is odd (k = 2m+1 for m >= 0), then 2^k = 2^(2m+1) = 2*(2^m)^2.  If k is even (k = 2m for m >= 0), then 2^k = 2^(2m) = (2^m)^2.  So, the powers of 2 sequence (A000079) is a subsequence of this one. - Timothy L. Tiffin, Jul 18 2016
Numbers n such that A175317(n) = Sum_{d|n} pod(d) is odd, where pod(m) = the product of divisors of m (A007955). - Jaroslav Krizek, Dec 28 2016
Positions of zeros in A292377 and A292383, positions of ones in A286357 and A292583. (See A292583 for why.) - Antti Karttunen, Sep 25 2017
Numbers of the form A000079(i)*A016754(j), i,j>=0. - R. J. Mathar, May 30 2020
Equivalently, numbers whose odd part is square. Cf. A042968. - Peter Munn, Jul 14 2020
These are the Heinz numbers of the partitions counted by A119620. - Gus Wiseman, Oct 29 2021
Numbers m whose abundance, A033880(m), is odd. - Peter Munn, May 23 2022
Numbers with an odd number of middle divisors (cf. A067742). - Omar E. Pol, Aug 02 2022

T. D. Noe, Table of n, a(n) for n = 1..1000
Tewodros Amdeberhan, Victor H. Moll, Vaishavi Sharma, and Diego Villamizar, Arithmetic properties of the sum of divisors, arXiv:2007.03088 [math.NT], 2020. See p. 5.
J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, Journal of Integer Sequences, Vol. 16 (2013), #13.1.8.
Patrick De Geest, World!Of Numbers
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163.
Eric Weisstein's World of Mathematics, Abundance

Complement of A028983.
Characteristic function is A053866, A093709.
Odd terms in A178910.
Cf. A000203, A000290, A000593, A001105, A042968, A187793.
Supersequence of A000079.
Cf. A028260, A033880, A046951, A067742.
Cf. A119620, A119899, A347437, A347438, A348550.

import Data.List.Ordered (union)
a028982 n = a028982_list !! (n-1)
a028982_list = tail $ union a000290_list a001105_list
-- Reinhard Zumkeller, Jun 27 2015

Take[ Sort[ Flatten[ Table[{n^2, 2n^2}, {n, 35}] ]], 57] (* Robert G. Wilson v, Aug 27 2004 *)

list(lim)=vecsort(concat(vector(sqrtint(lim\1),i,i^2), vector(sqrtint(lim\2),i,2*i^2))) \\ Charles R Greathouse IV, Jun 16 2011

from itertools import count, islice
from sympy.ntheory.primetest import is_square
def A028982_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:int(is_square(n) or is_square(n<<1)),count(max(startvalue,1)))
A028982_list = list(islice(A028982_gen(),30)) # Chai Wah Wu, Jan 09 2023

from math import isqrt
def A028982(n):
    def f(x): return n-1+x-isqrt(x)-isqrt(x>>1)
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 22 2024

A001105 UNION A000290.
a(n) is asymptotic to c*n^2 with c = 2/(1+sqrt(2))^2 = 0.3431457.... - Benoit Cloitre, Sep 17 2002
In particular, a(n) = c*n^2 + O(n). - Charles R Greathouse IV, Jan 11 2013
a(A003152(n)) = n^2; a(A003151(n)) = 2*n^2. - Enrique Pérez Herrero, Oct 09 2013
Sum_{n>=1} 1/a(n) = Pi^2/4. - Amiram Eldar, Jun 28 2020

A266265 Product of products of divisors of divisors of n.

Original entry on oeis.org

1, 2, 3, 16, 5, 216, 7, 1024, 81, 1000, 11, 2985984, 13, 2744, 3375, 1048576, 17, 34012224, 19, 64000000, 9261, 10648, 23, 63403380965376, 625, 17576, 59049, 481890304, 29, 19683000000000, 31, 34359738368, 35937, 39304, 42875, 4738381338321616896, 37, 54872

Offset: 1

Jaroslav Krizek, Dec 25 2015

nonn

a(n) = Product_{d|n} A007955(d) where A007955(m) = product of divisors of m.
From G. C. Greubel, Dec 31 2015: (Start)
for n>=1: 10^3|a(10*n), 10^6|a(20*n), 10^9|a(30*n).
for n>=0: 10^6|a(60*n+50), 10^9|a(60*n+70). (End)

			For n = 6; with b(n) = A007955(n); a(6) = b(1)*b(2)*b(3)*b(6) = 1*2*3*36 = 216.

G. C. Greubel, Table of n, a(n) for n = 1..1200

Cf. A007955 (product of divisors of n), A175317 (sum of products of divisors of divisors of n), A206032 (product of sums of divisors of divisors of n).

[&*[&*[b: b in Divisors(d)]: d in Divisors(n)]: n in [1..100]]

A266265 := proc(n)
    mul( A007955(d),d=numtheory[divisors](n)) ;
end proc:
seq(A266265(n),n=1..10) ; # R. J. Mathar, Feb 13 2019

Table[Product[Times @@ Divisors@ d, {d, Divisors@ n}], {n, 38}] (* Michael De Vlieger, Dec 31 2015 *)

a(n) = {my(d = divisors(n)); prod(k=1, #d, dd = divisors(d[k]); prod(kk=1,#dd, dd[kk]));} \\ Michel Marcus, Dec 27 2015

a(p) = p for p = prime.

a(n) = Product_{d|n} d^tau(n/d). - Ridouane Oudra, Apr 09 2023

A280116 Partial sums of A266265 (Product_{d|n} pod(d)).

Original entry on oeis.org

1, 3, 6, 22, 27, 243, 250, 1274, 1355, 2355, 2366, 2988350, 2988363, 2991107, 2994482, 4043058, 4043075, 38055299, 38055318, 102055318, 102064579, 102075227, 102075250, 63403483040626, 63403483041251, 63403483058827, 63403483117876, 63403965008180

Offset: 1

Jaroslav Krizek, Dec 26 2016

nonn

pod(n) is the product of the divisors of n (A007955).

Cf. A007955, A175317, A266265, A280114 (partial sums of A175317), A280117 (partial products of A266265).

[&+[&*[&*[b: b in Divisors(d)]: d in Divisors(k)]: k in [1..n]]: n in [1..1000]];

Accumulate@ Array[Product[Times @@ Divisors@ d, {d, Divisors@ #}] &, 28] (* Michael De Vlieger, Dec 26 2016 *)

a(n) = Sum_{i=1..n} A266265(i).

cp's OEIS Frontend

A280114 Partial sums of A175317 (Sum_{d|n} pod(d)).

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

Formula

A280115 Partial products of A175317 (Sum_{d|n} pod(d)).

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

Formula

A028982 Squares and twice squares.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

Python

Formula

A266265 Product of products of divisors of divisors of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A280116 Partial sums of A266265 (Product_{d|n} pod(d)).

Views

Author

Keywords

Comments

Crossrefs

Programs

Magma

Mathematica

Formula

A280117 Partial products of A266265 (Product_{d|n} pod(d)), where pod(n) is the product of the divisors of n (A007955).

Views

Author

Keywords

Crossrefs

Programs

Magma

Formula