A069934 -id:A069934 - cp's OEIS frontend

A334471 a(n) = Product_{d|n} (A069934(n) / sigma(d)) where A069934(n) = lcm_{d|n} sigma(d).

1, 3, 4, 441, 6, 144, 8, 385875, 2704, 324, 12, 12446784, 14, 576, 576, 37418184916875, 18, 197413632, 20, 42007896, 1024, 1296, 24, 38118276000000, 34596, 1764, 35152000, 99574272, 30, 26873856, 32, 1409355934894096875, 2304, 2916, 2304, 1695648500686393344

Offset: 1

Jaroslav Krizek, May 01 2020

nonn

			For n = 6; divisors d of 6: {1, 2, 3, 6}; sigma(d): {1, 3, 4, 12}; lcm_{d|6} sigma(d) = 12; a(6) = 12/1 * 12/3 * 12/4 * 12/12 = 144.

Cf. Similar sequence with tau(d): A334470.

Cf. A000005, A000040, A069934, A206032.

[&*[ LCM([&+Divisors(d): d in Divisors(n)]) / &+Divisors(d): d in Divisors(n)]: n in [1..100]]

a[n_] := (LCM @@ (s = DivisorSigma[1, Divisors[n]]))^Length[s] / Times @@ s; Array[a, 36] (* Amiram Eldar, May 02 2020 *)

a(n) = {my(d=divisors(n), lcms = lcm(vector(#d, k, sigma(d[k])))); vecprod(vector(#d, k, lcms/sigma(d[k])));} \\ Michel Marcus, May 02 2020

a(n) = ((lcm_{d|n} sigma(d))^tau(n)) / Product_{d|n} (sigma(d)).
a(n) = A069934(n)^A000005(n) / A206032(n).
a(p) = p + 1 for p = primes (A000040).

A265709 a(n) = numerator of Sum_{d|n} 1/sigma(d).

Original entry on oeis.org

1, 4, 5, 31, 7, 5, 9, 54, 69, 14, 13, 155, 15, 3, 35, 1709, 19, 23, 21, 31, 45, 13, 25, 27, 223, 10, 703, 93, 31, 35, 33, 15536, 65, 38, 21, 713, 39, 7, 75, 9, 43, 15, 45, 403, 161, 25, 49, 1709, 521, 446, 95, 155, 55, 703, 91, 243, 21, 62, 61, 155, 63, 11

Offset: 1

Jaroslav Krizek, Dec 24 2015

a(n) = numerator of Sum_{d|n} 1/A000203(d).

Are there numbers n > 1 such that Sum_{d|n} 1/sigma(d) is an integer?

			For n = 6; divisors d of 6: {1, 2, 3, 6}; sigma(d): {1, 3, 4, 12}; Sum_{d|6} 1/sigma(d) = 1/1 + 1/3 + 1/4 + 1/12 = 20/12 = 5/3; a(n) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A069934, A000203, A265708, A265710, A265711, A265712, A265713, A265714, A266227, A266228.

[Numerator(&+[1/SumOfDivisors(d): d in Divisors(n)]): n in [1..1000]]

A265709[n_] := Numerator[DivisorSum[n, 1/DivisorSigma[1,#]&]];
Array[A265709, 100] (* Paolo Xausa, Feb 06 2024 *)

A265709(n) = numerator(sumdiv(n,d,1/sigma(d))); \\ Antti Karttunen, Nov 19 2017

a(n) = A265710(n) * Sum_{d|n} 1/sigma(d) = A265708(n) * A265710(n) / A069934(n).

a(1) = 1; a(p) = p + 2 for p = prime.

A265710 a(n) = denominator of Sum_{d|n} 1/sigma(d).

Original entry on oeis.org

1, 3, 4, 21, 6, 3, 8, 35, 52, 9, 12, 84, 14, 2, 24, 1085, 18, 13, 20, 18, 32, 9, 24, 14, 186, 7, 520, 56, 30, 18, 32, 9765, 48, 27, 16, 364, 38, 5, 56, 5, 42, 8, 44, 252, 104, 18, 48, 868, 456, 279, 72, 98, 54, 390, 72, 140, 16, 45, 60, 72, 62, 8, 416, 1240155

Offset: 1

Jaroslav Krizek, Dec 24 2015

a(n) = denominator of Sum_{d|n} 1/A000203(d).
Are there numbers n > 1 such that Sum_{d|n} 1/sigma(d) is an integer?
a(n) = 2 for n = 14, 244, 494, 45994.  Are there any others? - Robert Israel, Apr 02 2017

			For n = 6; divisors d of 6: {1, 2, 3, 6}; sigma(d): {1, 3, 4, 12}; Sum_{d|6} 1/sigma(d) = 1/1 + 1/3 + 1/4 + 1/12 = 20/12 = 5/3; a(n) = 3.

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

Cf. A069934, A000203, A265708, A265709, A265711, A265712, A265713, A265714, A266227, A266228.

f:= n -> denom(add(1/numtheory:-sigma(d), d = numtheory:-divisors(n))):
map(f, [$1..200]); # Robert Israel, Apr 02 2017

Table[Denominator[Plus@@(1/DivisorSigma[1, Divisors[n]])], {n, 70}] (* Alonso del Arte, Dec 24 2015 *)

a(n) = denominator(sumdiv(n, d, 1/sigma(d))); \\ Michel Marcus, Feb 06 2024

a(1) = 1; a(p) = p + 1 for p = prime.

a(n) = A265709(n) / (Sum_{d|n} 1/sigma(d)) = A265709(n) * A069934(n) / A265708(n).

A265708 a(n) = lcm_{d|n} sigma(d) * Sum_{d|n} 1/sigma(d), where sigma(d) represents the sum of divisors of d (A000203(d)).

Original entry on oeis.org

1, 4, 5, 31, 7, 20, 9, 162, 69, 28, 13, 155, 15, 36, 35, 5127, 19, 276, 21, 217, 45, 52, 25, 810, 223, 60, 703, 279, 31, 140, 33, 15536, 65, 76, 63, 2139, 39, 84, 75, 1134, 43, 180, 45, 403, 483, 100, 49, 25635, 521, 892, 95, 465, 55, 2812, 91, 1458, 105, 124

Offset: 1

Jaroslav Krizek, Dec 24 2015

			For n = 6; divisors d of 6: {1, 2, 3, 6}; sigma(d): {1, 3, 4, 12}; lcm_{d|6} sigma(d) = 12; a(6) = 12/1 + 12/3 + 12/4 + 12/12 = 20.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A069934, A000203, A265708, A265709, A265710, A265711, A265712, A265713, A265714, A266227, A266228.

[&+[1/SumOfDivisors(d): d in Divisors(n)] * LCM([SumOfDivisors(d): d in Divisors(n)]): n in [1..100]];

a[n_] := LCM @@ DivisorSigma[1, Divisors[n]] * DivisorSum[n, 1/DivisorSigma[1, #] &]; Array[a, 100] (* Amiram Eldar, Dec 09 2022 *)

A069934(n) = my(d = divisors(n)); lcm(vector(#d, k, sigma(d[k])));
A265708(n) = (A069934(n) * sumdiv(n,d,1/sigma(d))); \\ Antti Karttunen, Nov 19 2017

a(n) = A069934(n) * Sum_{d|n} 1/A000203(d) = A265709(n) * A069934(n) / A265710(n).

Multiplicative with a(p^e) = (1/1 + ..., + 1/sigma(p^(e-1)) + 1/sigma(p^(e))) * lcm{1, ..., sigma(p^(e-1)), sigma(p^(e))}.

A265711 Numbers n such that floor(Sum_{d|n} 1 / sigma(d)) = 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 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, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73

Offset: 1

Jaroslav Krizek, Dec 25 2015

nonn

Numbers n such that A265710(n) = floor(A265708(n) / A069934(n)) = floor(A265709(n) / A265710(n)) = 1.

See A265714(n) = the smallest number k such that floor(Sum_{d|k} 1/sigma(d)) = n.

			6 is a term because floor(Sum_{d|6} 1/sigma(d)) = floor(1/1 + 1/3 + 1/4 + 1/12) = floor(5/3) = 1.

G. C. Greubel, Table of n, a(n) for n = 1..9268

Cf. A069934, A000203, A265708, A265709, A265710, A265712, A265713, A265714, A266227, A266228.

[n: n in [1..1000] | Floor(&+[1/SumOfDivisors(d): d in Divisors(n)]) eq 1]

Select[Range@ 73, Floor[Sum[1/DivisorSigma[1, d], {d, Divisors@ #}]] == 1 &] (* Michael De Vlieger, Dec 31 2015 *)

isok(n) = floor(sumdiv(n, d, 1/sigma(d))) == 1; \\ Michel Marcus, Dec 27 2015

A265712 Numbers n such that floor(Sum_{d|n} 1 / sigma(d)) = 2.

Original entry on oeis.org

60, 72, 84, 90, 120, 144, 168, 180, 210, 216, 240, 252, 264, 270, 280, 288, 300, 312, 324, 330, 336, 360, 378, 384, 390, 396, 408, 420, 432, 450, 456, 462, 468, 480, 504, 510, 528, 540, 546, 552, 560, 570, 576, 588, 600, 612, 624, 630, 648, 660, 672, 684, 690

Offset: 1

Jaroslav Krizek, Dec 25 2015

nonn

Numbers n such that A265710(n) = floor(A265708(n) / A069934(n)) = floor(A265709(n) / A265710(n)) = 2.

See A265714(n) = the smallest number k such that floor(Sum_{d|k} 1/sigma(d)) = n.

			60 is a term because floor(Sum_{d|60} 1/sigma(d)) = floor(155/72) = 2.

G. C. Greubel, Table of n, a(n) for n = 1..7334

Cf. A069934, A000203, A265708, A265709, A265710, A265711, A265713, A265714, A266227, A266228.

[n: n in [1..1000] | Floor(&+[1/SumOfDivisors(d): d in Divisors(n)]) eq 2]

Select[Range@ 690, Floor[Sum[1/DivisorSigma[1, d], {d, Divisors@ #}]] == 2 &] (* Michael De Vlieger, Dec 31 2015 *)

isok(n) = floor(sumdiv(n, d, 1/sigma(d))) == 2; \\ Michel Marcus, Dec 27 2015

A265713 Numbers k such that floor(Sum_{d|k} 1 / sigma(d)) = 3.

Original entry on oeis.org

110880, 166320, 221760, 277200, 327600, 332640, 360360, 388080, 393120, 415800, 443520, 471240, 480480, 491400, 498960, 526680, 540540, 554400, 556920, 582120, 589680, 600600, 622440, 637560, 655200, 665280, 693000, 720720, 776160, 786240, 803880, 831600

Offset: 1

Jaroslav Krizek, Dec 25 2015

nonn

Numbers k such that A265710(k) = floor(A265708(k) / A069934(k)) = floor(A265709(k) / A265710(k)) = 3.

See A265714(n) = the smallest number k such that floor(Sum_{d|k} 1/sigma(d)) = n.

			110880 is a term because floor(Sum_{d|110880} 1/sigma(d)) = floor(22333/7440) = 3.

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

Cf. A069934, A000203, A265708, A265709, A265710, A265711, A265712, A265714, A266227, A266228.

[n: n in [1..1000000] | Floor(&+[1/SumOfDivisors(d): d in Divisors(n)]) eq 3]

Select[Range[10^5, 9*10^5], Floor[Sum[1/DivisorSigma[1, d], {d, Divisors@ #}]] == 3 &] (* Michael De Vlieger, Dec 31 2015 *)

isok(n) = floor(sumdiv(n, d, 1/sigma(d))) == 3; \\ Michel Marcus, Dec 27 2015

A266227 a(n) = floor(Sum_{d|n} 1/sigma(d)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Jaroslav Krizek, Dec 24 2015

nonn

Sequence of numbers n such that floor(Sum_{d|n} 1/sigma(d)) = k for k = 1, 2, 3:
k = 1: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ... (A265711);
k = 2: 60, 72, 84, 90, 120, 144, 168, 180, 210, 216, 240, 252, ... (A265712);
k = 3: 110880, 166320, 221760, 277200, 327600, 332640, 360360, ... (A265713).

			For n = 6; a(6) = floor(Sum_{d|6} 1/sigma(d)) = floor(1/1 + 1/3 + 1/4 + 1/12) = floor(5/3) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A069934, A000203, A265708, A265709, A265710, A265711, A265712, A265713, A265714, A266228.

[Floor(&+[1/SumOfDivisors(d): d in Divisors(n)]): n in [1..100]]

A266227[n_] := Floor[DivisorSum[n, 1/DivisorSigma[1, #]&]];
Array[A266227, 100] (* Paolo Xausa, Feb 06 2024 *)

A266227(n) = { my(s=sumdiv(n,d,1/sigma(d))); (numerator(s) \ denominator(s)); }; \\ Antti Karttunen, Nov 19 2017

a(n) = floor(Sum_{d|n} 1/A000203(d)).

a(n) = floor(A265708(n) / A069934(n)) = floor(A265709(n) / A265710(n)).

A265714 a(n) = the smallest number k such that floor(Sum_{d|k} 1/sigma(d)) = n.

Original entry on oeis.org

1, 60, 110880, 4658179125600, 950542574818669103079134726400, 204614292026733833316841991529248485168966921782532186656980932752000

Offset: 1

Jaroslav Krizek, Dec 24 2015

nonn

If a(4) exists, it must be bigger than 5*10^6.
Are there numbers n > 1 such that Sum_{d|n} 1/sigma(d) is an integer?
Conjecture: a(n) is also the smallest number k such that Sum_{d|k} 1/sigma(d) >= n.
Sequence of numbers n such that floor(Sum_{d|n} 1/sigma(d)) = k for k = 1, 2, 3:
k = 1: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ... (A265711);
k = 2: 60, 72, 84, 90, 120, 144, 168, 180, 210, 216, 240, 252, ... (A265712);
k = 3: 110880, 166320, 221760, 277200, 327600, 332640, 360360, ... (A265713).
From Robert Israel, Dec 24 2015: (Start)
Note that g(k) = Sum_{d | k} 1/sigma(d) is multiplicative, with g(p) = 1 + 1/(p+1) for prime p.
Since Product_p (1+1/(p+1)) diverges, there are certainly numbers k with floor(g(k)) = n, including some squarefree numbers.
Conjectured values: a(4) = 4658179125600,
  a(5) = 1188178218523336378848918408000,
  a(6) = 5354073974699535305124032111682002028587967786642925550857667740344000.
These do have the correct value of floor(g(k)), but may not be the lowest possible.
(End)
Probably, a(7) = 1058687979...2471360000 = 349# * 23# * 7# * 5# * 3#^2 * 2#^3. - Hiroaki Yamanouchi, Dec 31 2015

			For n = 2; a(2) = 60 because 60 is the smallest number with floor (Sum_{d|60} 1/sigma(d)) = floor(155/72) = 2.

Cf. A069934, A000203, A265708, A265709, A265710, A265711, A265712, A265713, A266227, A266228.

a:=1; S:=[a]; for n in [2..3] do k:=0; flag:= true; while flag do k+:=1; if &+[1/SumOfDivisors(d): d in Divisors(k)] ge n then Append(~S, k); a:=k; flag:=false; end if; end while; end for; S;

a(4)-a(6) from Hiroaki Yamanouchi, Dec 31 2015

A266228 Numbers n such that Sum_{d|n} 1/sigma(d) > Sum_{d|m} 1/sigma(d) for all m < n.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 30, 36, 48, 60, 120, 180, 240, 360, 420, 720, 840, 1260, 1680, 2520, 5040, 10080, 13860, 15120, 18480, 25200, 27720, 55440, 110880, 166320, 277200, 332640, 360360, 720720, 1441440, 2162160, 3603600, 4324320, 6126120, 7207200, 10810800, 12252240, 21621600, 24504480, 36756720, 61261200, 73513440

Offset: 1

Jaroslav Krizek, Dec 24 2015

nonn

Where record values of Sum_{d|n} 1/sigma(d) occur.

			For n = 4; a(4) = 6 because 6 is the smallest number such that Sum_{d|a(4)} 1/sigma(d) = Sum_{d|6} 1/sigma(d) = 5/3 > Sum_{d|a(3)} 1/sigma(d) = Sum_{d|4} 1/sigma(d) = 31/21.

Cf. A069934, A000203, A265708, A265709, A265710, A265711, A265712, A265713, A265714, A266227.

a:=1; S:=[a]; for n in [2..25] do k:=0; flag:= true; while flag do k+:=1; if &+[1/SumOfDivisors(d): d in Divisors(a)] lt &+[1/SumOfDivisors(d): d in Divisors(k)] then Append(~S, k); a:=k; flag:=false; end if; end while; end for; S;

b(n)={sumdiv(n, d, 1/sigma(d));}
{ my(m=0); for(n=1, 1e6, if(b(n)>m, m=b(n); print1(n, ", "))) } \\ Andrew Howroyd, Nov 11 2018

a(35)-a(47) from Andrew Howroyd, Nov 11 2018

cp's OEIS Frontend

A334471 a(n) = Product_{d|n} (A069934(n) / sigma(d)) where A069934(n) = lcm_{d|n} sigma(d).

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A265709 a(n) = numerator of Sum_{d|n} 1/sigma(d).

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A265710 a(n) = denominator of Sum_{d|n} 1/sigma(d).

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A265708 a(n) = lcm_{d|n} sigma(d) * Sum_{d|n} 1/sigma(d), where sigma(d) represents the sum of divisors of d (A000203(d)).

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A265711 Numbers n such that floor(Sum_{d|n} 1 / sigma(d)) = 1.

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Magma

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A265712 Numbers n such that floor(Sum_{d|n} 1 / sigma(d)) = 2.

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A265713 Numbers k such that floor(Sum_{d|k} 1 / sigma(d)) = 3.

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