A050999 -id:A050999 - cp's OEIS frontend

A363969 Expansion of Sum_{k>0} k^2 * x^(2k-1) / (1 - x^(2k-1)).

1, 1, 5, 1, 10, 5, 17, 1, 30, 10, 37, 5, 50, 17, 78, 1, 82, 30, 101, 10, 142, 37, 145, 5, 179, 50, 226, 17, 226, 78, 257, 1, 330, 82, 350, 30, 362, 101, 454, 10, 442, 142, 485, 37, 632, 145, 577, 5, 642, 179, 762, 50, 730, 226, 830, 17, 946, 226, 901, 78, 962, 257, 1191, 1, 1148, 330, 1157, 82

Offset: 1

Seiichi Manyama, Jun 30 2023

Cf. A001227, A113415.

Cf. A050999, A363974.

a[n_] := DivisorSum[n, ((# + 1)/2)^2 &, OddQ[#] &]; Array[a, 100] (* Amiram Eldar, Jun 30 2023 *)

a(n) = sumdiv(n, d, (d%2==1)*((d+1)/2)^2);

a(n) = Sum_{d|n, d==1 mod 2} ((d+1)/2)^2.

A374539 The sum of the squares of the infinitary divisors of n.

Original entry on oeis.org

1, 5, 10, 17, 26, 50, 50, 85, 82, 130, 122, 170, 170, 250, 260, 257, 290, 410, 362, 442, 500, 610, 530, 850, 626, 850, 820, 850, 842, 1300, 962, 1285, 1220, 1450, 1300, 1394, 1370, 1810, 1700, 2210, 1682, 2500, 1850, 2074, 2132, 2650, 2210, 2570, 2402, 3130, 2900

Offset: 1

Amiram Eldar, Jul 11 2024

Also the sum of the infinitary divisors of n^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A049417, A050376, A077609.

Similar sequences: A001157, A034676, A050999, A067558, A076577, A351265, A351307, A351647.

f[p_, e_] := p^(2^Position[Reverse@IntegerDigits[e, 2], ?(# == 1 &)]); a[1] = 1; a[n] := Times @@ (Flatten@(f @@@ FactorInteger[n]) + 1); Array[a, 100]

a(n) = {my(f = factor(n), b); prod(i = 1, #f~, b = binary(2 * f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)));}

from math import prod
from sympy import factorint
def A374539(n): return prod(p**(1<Chai Wah Wu, Jul 11 2024

a(n) = A049417(n^2).
a(n) <= A001157(n), with equality if and only if n is in A036537.
Multiplicative with a(p^e) = Product{k>=1, e_k=1} (p^(2^(k+1)) + 1), where e = Sum_{k} e_k * 2^k is the binary representation of e, i.e., e_k is bit k of e.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Product_{P} (1 + 1/(P^2*(P+1))) = 1.14142906130350119631..., and P are numbers of the form p^(2^k) where p is prime and k >= 0 (A050376).

A380742 Even numbers m such that the sum of the squares of the odd divisors and the sum of the squares of even divisors of m are both squares.

Original entry on oeis.org

2, 574, 3346, 12474, 19598, 19710, 42770, 73062, 93310, 133630, 250510, 365330, 425898, 485758, 546530, 761022, 782690, 1254430, 1460290, 1628926, 2139790, 2174018, 2286954, 2332798, 2845154, 3185870, 3630146, 4562510, 5089394, 5444010, 5656770, 6265870, 6377618

Offset: 1

Michel Lagneau, Jan 31 2025

nonn

Let s2 = 4*A001157(m/2) be the sum of the squares of the even divisors of m and s1 = A050999(m) be the sum of the squares of the odd divisors of m. We observe that s2/s1 = 4.

			574 is in the sequence because: its divisors are {1, 2, 7, 14, 41, 82, 287, 574}; the sum of squares of the odd divisors is 84100 which is square, and the sum of squares of the even divisors is 336400 which is square.

Cf. A000203, A000593, A001157, A050999, A146076.

with(numtheory):nn:=10^7:
for m from 2 by 2 to nn do:
 d:=divisors(m):
   s1:=0: s2:=0:
   for i in d do
    if i::odd then s1:=s1+i^2 else s2:=s2+i^2 fi
   od:
   if issqr(s2) and issqr(s1) then print(m) fi
 od:

Select[Range[2,10^6,2],AllTrue[{Sqrt[Total[Select[Divisors[#],OddQ]^2]],Sqrt[Total[Select[Divisors[#],EvenQ]^2]]},IntegerQ]&] (* James C. McMahon, Feb 10 2025 *)

isok(k) = !(k%2) && issquare(sumdiv(k, d, if (d%2, d^2))) && issquare(sumdiv(k, d, if (1-d%2, d^2))); \\ Michel Marcus, Feb 22 2025

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A363969 Expansion of Sum_{k>0} k^2 * x^(2k-1) / (1 - x^(2k-1)).

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A374539 The sum of the squares of the infinitary divisors of n.

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A380742 Even numbers m such that the sum of the squares of the odd divisors and the sum of the squares of even divisors of m are both squares.

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cp's OEIS Frontend

A363969 Expansion of Sum_{k>0} k^2 * x^(2*k-1) / (1 - x^(2*k-1)).

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A374539 The sum of the squares of the infinitary divisors of n.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A380742 Even numbers m such that the sum of the squares of the odd divisors and the sum of the squares of even divisors of m are both squares.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

A363969 Expansion of Sum_{k>0} k^2 * x^(2k-1) / (1 - x^(2k-1)).