A060780 -id:A060780 - cp's OEIS frontend

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

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

A083539 a(n) = sigma(n) * sigma(n+1): product of sigma-values for consecutive integers.

Original entry on oeis.org

3, 12, 28, 42, 72, 96, 120, 195, 234, 216, 336, 392, 336, 576, 744, 558, 702, 780, 840, 1344, 1152, 864, 1440, 1860, 1302, 1680, 2240, 1680, 2160, 2304, 2016, 3024, 2592, 2592, 4368, 3458, 2280, 3360, 5040, 3780, 4032, 4224, 3696, 6552, 5616, 3456, 5952

Offset: 1

Labos Elemer, May 21 2003

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000203, A002378, A060781, A060780, A083538, A330322 (partial sums).

Cf. A083481, A083542, A092517.

f[x_] := DivisorSigma[1, x]; t=Table[f[w+1]*f[w], {w, 1, 128}]
Times@@@Partition[DivisorSigma[1,Range[50]],2,1] (* Harvey P. Dale, May 21 2014 *)

a(n)=sigma(n)*sigma(n+1) \\ Charles R Greathouse IV, Feb 14 2013

a(n) = A000203(A002378(n)). - Amiram Eldar, Jul 10 2024

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

Original entry on oeis.org

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 *)

A084307 a(n) is the least number x such that gcd(sigma(x), sigma(x+1)) = n.

Original entry on oeis.org

1, 13, 17, 6, 199, 5, 242, 27, 391, 57, 1296, 22, 882, 12, 648, 93, 175232, 89, 3872, 236, 195, 1032, 4875263, 14, 5739271, 467, 35377, 83, 1882384, 58, 3024, 308, 29240, 201, 1627208, 118, 79524, 147, 1682, 56, 46854024, 82, 229441, 1204, 2047, 6301, 83386957823

Offset: 1

Labos Elemer, Jun 13 2003

nonn

a(47) > 10^9 if it exists. - Michel Marcus, Sep 01 2019

			n=9: GCD[sigma[x+1], sigma[x]]=5 holds for {391,799,800,881,...} of which the first is a(9)=391.

Cf. A000203, A060780, A063444.

f[x_] := GCD[DivisorSigma[1, x], DivisorSigma[1, x+1]] t=Table[0, {256}]; Do[s=f[n]; If[s<257&&t[[s]]==0, t[[s]]=n], {n, 1, 10000000}]; t

a(n) = my(x=1, sx=sigma(x), y=2, sy=sigma(2)); while(gcd(sx, sy) != n, x++; y++; sx=sy; sy=sigma(y)); x; \\ Michel Marcus, Aug 28 2019

a(41) from Rémy Sigrist, Aug 29 2019
a(42)-a(46) from Michel Marcus, Aug 30 2019
a(47) from Giovanni Resta, Oct 29 2019

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))).

A070010 GCD of consecutive values of sum-of-proper divisors.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 11, 1, 1, 1, 6, 2, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 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, 4, 1, 1, 1, 23, 1, 1, 1, 1, 3, 1, 1, 5, 25, 1, 1, 1, 1, 3, 1, 1, 1, 1

Offset: 1

Labos Elemer, Apr 11 2002

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000203, A001065; GCD of various consecutive function values: A048586, A057467, A058515, A060778, A060780, A069896.

GCD[DivisorSigma[1, n+1]-(n+1), DivisorSigma[1, n]-n]

A001065(n) = (sigma(n) - n);
A070010(n) = gcd(A001065(n),A001065(1+n)); \\ Antti Karttunen, Oct 05 2017

a(n) = gcd(A001065(n+1), A001065(n)).

A322569 a(n)=x is the least integer such that gcd(sigma(x), sigma(x+1)) = 2*n.

Original entry on oeis.org

13, 6, 5, 27, 57, 22, 12, 93, 89, 236, 1032, 14, 467, 83, 58, 308, 201, 118, 147, 56, 82, 1204, 6301, 69, 596, 1142, 106, 91, 4167, 87, 432, 381, 393, 1407, 348, 70, 5912, 453, 233, 417, 13692, 166, 56493, 1118, 88, 6987, 54048, 154, 1843, 4490, 6833, 2574, 633, 689, 1538

Offset: 1

Michel Marcus, Aug 29 2019

nonn

Bisection of A084307.

Michel Marcus, Table of n, a(n) for n = 1..1000

Cf. A000203, A060780, A084307.

sol:=[]; for n in [1..55] do k:=1; while Gcd(DivisorSigma(1,k),DivisorSigma(1,k+1)) ne 2*n do k:=k+1; end while; Append(~sol,k); end for; sol; // Marius A. Burtea, Aug 29 2019

Module[{nn=60000,g},g=GCD@@@Partition[DivisorSigma[1,Range[nn]],2,1];Table[ Position[ g,2n,1,1],{n,55}]]//Flatten (* Harvey P. Dale, Jan 28 2023 *)

a(n) = my(x=1); while(gcd(sigma(x), sigma(x+1)) != 2*n, x++); x;

cp's OEIS Frontend

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

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A083539 a(n) = sigma(n) * sigma(n+1): product of sigma-values for consecutive integers.

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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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A084307 a(n) is the least number x such that gcd(sigma(x), sigma(x+1)) = n.

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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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A070010 GCD of consecutive values of sum-of-proper divisors.

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A322569 a(n)=x is the least integer such that gcd(sigma(x), sigma(x+1)) = 2*n.

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