A083539 -id:A083539 - cp's OEIS frontend

A083540 Numbers k such that A083539(k) is a square; solutions x to sigma(x+1)*sigma(x)=y^2 for some y.

14, 30, 51, 161, 186, 206, 223, 329, 552, 713, 759, 869, 957, 994, 995, 1248, 1334, 1364, 1634, 1715, 1819, 2093, 2133, 2584, 2685, 2820, 2821, 2974, 3115, 3145, 3485, 4212, 4308, 4312, 4364, 4408, 4649

Offset: 1

Labos Elemer, May 21 2003

nonn

			x=30: sigma(30)=72, sigma(31)=32, product = 72*32 = 256*9 = 24^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A000203, A083539, A060780, A060781, A083538.

Do[s=Sqrt[DivisorSigma[1, n+1]*DivisorSigma[1, n]]; If[IntegerQ[s], Print[n]], {n, 1, 5000}]
Flatten[Position[Times@@@Partition[DivisorSigma[1,Range[5000]],2,1], ?(IntegerQ[Sqrt[#]]&)]] (* _Harvey P. Dale, Mar 07 2016 *)

A083541 Values of y from solutions to sigma(x+1)*sigma(x)=y^2, where A083539(x) = y^2 is a square number.

Original entry on oeis.org

24, 48, 84, 264, 288, 312, 336, 576, 960, 1152, 1440, 1440, 1440, 1440, 1680, 2100, 2160, 2688, 2640, 3360, 3024, 3360, 3360, 4320, 4320, 5376, 4032, 4464, 5040, 4788, 6048, 7392, 6720, 6840, 7644, 6300, 7440, 7560, 7020, 10080, 10080, 8064, 10080

Offset: 1

Labos Elemer, May 21 2003

nonn

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

Cf. A000203, A083539, A083540, A083538, A060780, A060781.

Do[s=Sqrt[DivisorSigma[1, n+1]*DivisorSigma[1, n]]; If[IntegerQ[s], Print[s]], {n, 1, 5000}]

a(n) = sqrt(sigma(A083540(n)) * sigma(1+A083540(n))).

A083538 a(n) = sigma(n)*sigma(n+1)/gcd(sigma(n+1), sigma(n))^2.

Original entry on oeis.org

3, 12, 28, 42, 2, 6, 120, 195, 234, 6, 21, 2, 84, 1, 744, 558, 78, 780, 210, 336, 72, 6, 10, 1860, 1302, 420, 35, 420, 60, 36, 2016, 336, 72, 72, 4368, 3458, 570, 210, 1260, 105, 112, 264, 231, 182, 156, 6, 372, 7068, 589, 744, 1764, 1323, 180, 15, 15, 6, 72, 6, 70

Offset: 1

Labos Elemer, May 21 2003

nonn

a(n) = A060781(n)/A060780(n) = A083539(n)/A060780(n)^2; quotient when lcm(sigma(n+1), sigma(n)) is divided by gcd(sigma(n+1), sigma(n)).

			n=10: sigma(10)=18, sigma(11)=12, lcm(18, 12)=36, gcd(18, 12)=6, a(10) = 36/6 = 6.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000010, A066813, A058515, A000203, A060781, A060780, A083539.

f[x_] := DivisorSigma[1, x] t=Table[LCM[f[w+1], f[w]]/GCD[f[w+1], f[w]], {w, 1, 128}]
Times@@#/(GCD@@#)^2&/@Partition[DivisorSigma[1,Range[60]],2,1] (* Harvey P. Dale, Feb 17 2016 *)

a(n)=my(x=sigma(n),y=sigma(n+1)); x*y/gcd(x,y)^2 \\ Charles R Greathouse IV, Mar 09 2014

Edited by N. J. A. Sloane, Apr 29 2007

Corrections by Charles R Greathouse IV, Mar 09 2014

A092517 Product of tau values for consecutive integers.

Original entry on oeis.org

2, 4, 6, 6, 8, 8, 8, 12, 12, 8, 12, 12, 8, 16, 20, 10, 12, 12, 12, 24, 16, 8, 16, 24, 12, 16, 24, 12, 16, 16, 12, 24, 16, 16, 36, 18, 8, 16, 32, 16, 16, 16, 12, 36, 24, 8, 20, 30, 18, 24, 24, 12, 16, 32, 32, 32, 16, 8, 24, 24, 8, 24, 42, 28, 32, 16, 12, 24, 32, 16, 24, 24, 8, 24, 36

Offset: 1

Reinhard Zumkeller, Apr 06 2004

nonn

Number of divisors of the n-th oblong number. - Ray Chandler, Jun 23 2008
Number of positive solutions (x,y) for which n/x + (n+1)/y = 1. - Michel Lagneau, Jan 16 2014
Number of positive solutions for which 1/p + 1/q + 1/(p*q) = 1/n; set p=x and q=y-1 in the solutions (x,y) in the comment above. - Mo Li, Apr 27 2021
a(n) is the maximum number of b > 0, which allows us to write (n+1)^2 as a sum of n+1 parts. Each part is of the form b^c and c is an integer >= 0 independent for each part. For n = 2 this is 3^2 = 2^2 + 2^2 + 2^0 = 3^1 + 3^1 + 3^1 = 4^1 + 4^1 + 4^0 = 7^1 + 7^0 + 7^0, b = 2;3;4;7 and a(2) = 4. It is conjectured that for all n the number of possible b reaches a(n). - Thomas Scheuerle, Jan 12 2022

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A000005, A002378, A063123, A063440, A083539, A123000.

[ NumberOfDivisors(n^2+n) : n in [1..100]]; // Vincenzo Librandi, Apr 03 2011

with(numtheory): seq(tau(n)*tau(n+1),n=1..73); # Zerinvary Lajos, Jan 22 2007

Table[DivisorSigma[0,n^2+n],{n,100}] (* Giorgos Kalogeropoulos, Apr 28 2021 *)
Times@@#&/@Partition[DivisorSigma[0,Range[80]],2,1] (* Harvey P. Dale, Apr 21 2022 *)

a(n) = numdiv(n^2+n); \\ Michel Marcus, Jan 11 2020

from sympy import divisor_count
def A092517(n): return divisor_count(n)*divisor_count(n+1) # Chai Wah Wu, Jan 06 2022

a(n) = A000005(n)*A000005(n+1) = A000005(n*(n+1)) = A000005(A002378(n)) = 2*A063123(n).

Extended by Ray Chandler, Jun 23 2008

A083542 a(n) = phi(n+1)*phi(n), product of totients of two consecutive integers.

Original entry on oeis.org

1, 2, 4, 8, 8, 12, 24, 24, 24, 40, 40, 48, 72, 48, 64, 128, 96, 108, 144, 96, 120, 220, 176, 160, 240, 216, 216, 336, 224, 240, 480, 320, 320, 384, 288, 432, 648, 432, 384, 640, 480, 504, 840, 480, 528, 1012, 736, 672, 840, 640, 768, 1248, 936, 720, 960, 864, 1008

Offset: 1

Labos Elemer, May 21 2003

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000010, A002378, A058515, A065474, A066813, A330319 (partial sums).

Cf. A083481, A083539, A092517.

a083542 n = a000010 n * a000010 (n + 1)
a083542_list = zipWith (*) (tail a000010_list) a000010_list
-- Reinhard Zumkeller, Apr 22 2012

a:= n-> (p-> p(n)*p(n+1))(numtheory[phi]):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 21 2022

Times @@ EulerPhi@ # & /@ Partition[Range@ 58, 2, 1] (* Michael De Vlieger, Mar 25 2017 *)
Times@@@Partition[EulerPhi[Range[60]],2,1] (* Harvey P. Dale, Oct 29 2019 *)

a(n) = eulerphi(n) * eulerphi(n+1); \\ Amiram Eldar, Jul 10 2024

a(n) = A000010(A002378(n)). - Amiram Eldar, Jul 10 2024
Sum_{k=1..n} a(k) = c * n^3 / 3 + O((n*log(n))^2), where c = Product_{p prime} (1 - 2/p^2) = 0.322634... (A065474). - Amiram Eldar, Dec 09 2024
a(n) = A058515(n)*A066813(n). - Amiram Eldar, May 07 2025

A216626 Square array read by antidiagonals, T(n,k) = sum_{c|n,d|k} lcm(c,d) for n>=1, k>=1.

Original entry on oeis.org

1, 3, 3, 4, 7, 4, 7, 12, 12, 7, 6, 15, 10, 15, 6, 12, 18, 28, 28, 18, 12, 8, 28, 24, 27, 24, 28, 8, 15, 24, 30, 42, 42, 30, 24, 15, 13, 31, 32, 60, 16, 60, 32, 31, 13, 18, 39, 60, 56, 72, 72, 56, 60, 39, 18, 12, 42, 28, 51, 48, 70, 48, 51, 28, 42, 12, 28, 36

Offset: 1

Peter Luschny, Sep 12 2012

T(n,n) = A064950(n) = sum_{d|n} d*tau(d^2).
T(n,1) = T(1,n) = A000203(n) = sigma(n).
T(n,2) = T(2,n) = A062731(n) = sigma(2*n).
T(n+1,n) = A083539(n) = sigma(n+1)*sigma(n).
T(prime(n),1) = A008864(n) = prime(n)+1.

			[-----1---2---3----4----5----6----7----8----9---10---11---12]
[ 1]  1,  3,  4,   7,   6,  12,   8,  15,  13,  18,  12,  28
[ 2]  3,  7, 12,  15,  18,  28,  24,  31,  39,  42,  36,  60
[ 3]  4, 12, 10,  28,  24,  30,  32,  60,  28,  72,  48,  70
[ 4]  7, 15, 28,  27,  42,  60,  56,  51,  91,  90,  84, 108
[ 5]  6, 18, 24,  42,  16,  72,  48,  90,  78,  48,  72, 168
[ 6] 12, 28, 30,  60,  72,  70,  96, 124,  84, 168, 144, 150
[ 7]  8, 24, 32,  56,  48,  96,  22, 120, 104, 144,  96, 224
[ 8] 15, 31, 60,  51,  90, 124, 120,  83, 195, 186, 180, 204
[ 9] 13, 39, 28,  91,  78,  84, 104, 195,  55, 234, 156, 196
[10] 18, 42, 72,  90,  48, 168, 144, 186, 234, 112, 216, 360
[11] 12, 36, 48,  84,  72, 144,  96, 180, 156, 216,  34, 336
[12] 28, 60, 70, 108, 168, 150, 224, 204, 196, 360, 336, 270
.
Displayed as a triangular array:
    1;
    3,  3;
    4,  7,  4;
    7, 12, 12,  7;
    6, 15, 10, 15,  6;
   12, 18, 28, 28, 18, 12;
    8, 28, 24, 27, 24, 28,  8;
   15, 24, 30, 42, 42, 30, 24, 15;
   13, 31, 32, 60, 16, 60, 32, 31, 13;

Alois P. Heinz, Antidiagonals n = 1..141, flattened

Cf. A216620, A216621, A216622, A216623, A216624, A216625, A216627.

with(numtheory):
T:= (n, k) -> add(add(ilcm(c, d), c=divisors(n)), d=divisors(k)):
seq (seq (T(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Sep 12 2012

T[n_, k_] := Sum[LCM[c, d], {c, Divisors[n]}, {d, Divisors[k]}]; Table[T[n-k+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 25 2014 *)

def A216626(n, k) :
    cp = cartesian_product([divisors(n), divisors(k)])
    return reduce(lambda x,y: x+y, map(lcm, cp))
for n in (1..12): [A216626(n,k) for k in (1..12)]

A330322 a(n) = Sum_{i=1..n} sigma(i)*sigma(i+1), where sigma(n) = A000203(n) is the sum of the divisors of n.

Original entry on oeis.org

3, 15, 43, 85, 157, 253, 373, 568, 802, 1018, 1354, 1746, 2082, 2658, 3402, 3960, 4662, 5442, 6282, 7626, 8778, 9642, 11082, 12942, 14244, 15924, 18164, 19844, 22004, 24308, 26324, 29348, 31940, 34532, 38900, 42358, 44638, 47998, 53038, 56818, 60850

Offset: 1

N. J. A. Sloane, Dec 12 2019

nonn

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 163.

Amiram Eldar, Table of n, a(n) for n = 1..10000
A. E. Ingham, Some asymptotic formulae in the theory of numbers, Journal of the London Mathematical Society, Vol. s1-2, No. 3 (1927), pp. 202-208.

Cf. A000203.

Partial sums of A083539.

f[n_] := DivisorSigma[1, n] * DivisorSigma[1, n + 1]; Accumulate @ Array[f, 100] (* Amiram Eldar, Mar 08 2020 *)
Accumulate[Table[DivisorSigma[1, n*(n + 1)], {n, 1, 50}]] (* Vaclav Kotesovec, Aug 18 2021 *)

a(n) ~ (5/6) * n^3. - Amiram Eldar, Mar 08 2020

A083549 Quotient if least common multiple (lcm) of cototient values of consecutive integers is divided by the greatest common divisor (gcd) of the same pair of consecutive numbers.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 4, 12, 2, 6, 8, 8, 8, 56, 56, 8, 12, 12, 12, 12, 12, 12, 16, 80, 70, 126, 144, 16, 22, 22, 16, 208, 234, 198, 264, 24, 20, 12, 40, 24, 30, 30, 24, 56, 56, 24, 32, 224, 210, 570, 532, 28, 36, 60, 480, 672, 70, 30, 44, 44, 32, 864, 864, 544, 782, 46, 36, 900

Offset: 1

Labos Elemer, May 22 2003

nonn

			n=33: cototient(33) = 33-20 = 13, cototient(34) = 34-16 = 18;
lcm(13,18) = 234, gcd(13,18) = 1, so a(34) = 234.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A051953, A083538, A083539, A083540, A083541, A083542, A083543, A083544, A083545, A083546, A083547, A083548, A083549, A083550, A083551, A083552, A083553, A083554, A083555, A049586.

f[x_] := x-EulerPhi[x]; Table[LCM[f[w+1], f[w]]/GCD[f[w+1], f[w]], {w, 69}]
(* Second program: *)
Map[Apply[LCM, #]/Apply[GCD, #] &@ Map[# - EulerPhi@ # &, #] &, Partition[Range[69], 2, 1]] (* Michael De Vlieger, Mar 17 2018 *)

a(n) = lcm(A051953(n), A051952(n+1))/gcd(A051953(n), A051952(n+1)) = lcm(cototient(n+1), cototient(n))/A049586(n).

cp's OEIS Frontend

A083540 Numbers k such that A083539(k) is a square; solutions x to sigma(x+1)*sigma(x)=y^2 for some y.

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A083541 Values of y from solutions to sigma(x+1)*sigma(x)=y^2, where A083539(x) = y^2 is a square number.

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A083538 a(n) = sigma(n)*sigma(n+1)/gcd(sigma(n+1), sigma(n))^2.

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A092517 Product of tau values for consecutive integers.

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A083542 a(n) = phi(n+1)*phi(n), product of totients of two consecutive integers.

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A216626 Square array read by antidiagonals, T(n,k) = sum_{c|n,d|k} lcm(c,d) for n>=1, k>=1.

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A330322 a(n) = Sum_{i=1..n} sigma(i)*sigma(i+1), where sigma(n) = A000203(n) is the sum of the divisors of n.

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