A083538 -id:A083538 - cp's OEIS frontend

A083543 Duplicate of A083538.

3, 12, 28, 42, 2, 6, 120, 195, 234, 6, 21, 2, 84, 1, 744, 558, 78, 780, 210, 336, 72, 6, 10

Offset: 1

dead

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

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

A083555 Quotient of LCM of prime(n+1)-1 and prime(n)-1 and GCD of the same two numbers.

Original entry on oeis.org

2, 2, 6, 15, 30, 12, 72, 99, 154, 210, 30, 90, 420, 483, 598, 754, 870, 110, 1155, 1260, 156, 1599, 1804, 132, 600, 2550, 2703, 2862, 756, 72, 4095, 4420, 4692, 5106, 5550, 650, 702, 6723, 7138, 7654, 8010, 342, 9120, 2352, 9702, 1155, 1295, 12543, 12882

Offset: 1

Labos Elemer, May 22 2003

nonn

			n=25: prime(25)=97, prime(26)=101; a(25) = lcm(96,100)/gcd(96,100) = 2400/4 = 600.

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

Cf. A006093, A083538..A083554, A051537, A058263.

P:= seq(ithprime(i),i=1..100):
seq(ilcm(P[i+1]-1,P[i]-1)/igcd(P[i+1]-1,P[i]-1),i=1..99); # Robert Israel, Jun 11 2017

f[x_] := Prime[x]-1 Table[LCM[f[w+1], f[w]]/GCD[f[w+1], f[w]], {w, 1, 128}]
(* Second program: *)
Table[Apply[LCM[#1, #2]/GCD[#1, #2] &, Prime[n + {1, 0}] - 1], {n, 49}] (* Michael De Vlieger, Jun 11 2017 *)

first(n)=my(v=vector(n),p=2,k,g); forprime(q=3,, g=gcd(p-1,q-1); v[k++]=(p-1)*(q-1)/g^2; p=q; if(k==n, break)); v \\ Charles R Greathouse IV, Jun 11 2017

a(n) = lcm(A006093(n+1), A006093(n))/gcd(A006093(n+1), A006093(n));
a(n) = A083554(n)/A058263(n).
a(n) = A051537(A006093(n+1), A006093(n)). - Robert Israel, Jun 11 2017

A083553 Product of prime(n+1)-1 and prime(n)-1.

Original entry on oeis.org

2, 8, 24, 60, 120, 192, 288, 396, 616, 840, 1080, 1440, 1680, 1932, 2392, 3016, 3480, 3960, 4620, 5040, 5616, 6396, 7216, 8448, 9600, 10200, 10812, 11448, 12096, 14112, 16380, 17680, 18768, 20424, 22200, 23400, 25272, 26892, 28552, 30616, 32040

Offset: 1

Labos Elemer, May 22 2003

nonn

The conductor of x*prime(n) + y*prime(n+1); that is, for all k >= a(n), there exist nonnegative integers x and y such that k = x*prime(n) + y*prime(n+1). - T. D. Noe, Sep 22 2004

			n=25: a(25) = (97-1)*(101-1) = 9600.

David Bressoud and Stan Wagon, A Course in Computational Number Theory, Key College Pub., 2000, p. 46.

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

Cf. A000040, A006093, A058263, A083538-A083555, A099407 (terms halved), A172042 [= A000010(a(n))], A256617.
One more than A037165.
Column 3 of A379010.

f[x_] := Prime[x]-1; Table[f[w+1]*f[w], {w, 1, 128}]

A083553(n) = ((prime(1+n)-1)*(prime(n)-1)); \\ Antti Karttunen, Dec 14 2024

a(n) = A006093(n+1)*A006093(n) = (prime(n+1)-1)*(prime(n)-1).
a(n) = A037165(n) + 1.
a(n) = 2*A099407(n). - Antti Karttunen, Dec 14 2024

A083545 Numbers k such that the geometric mean of the Euler totient function of k and k+1 is an integer.

Original entry on oeis.org

1, 3, 15, 19, 95, 104, 125, 164, 194, 255, 259, 341, 491, 495, 504, 512, 513, 584, 591, 629, 679, 755, 775, 975, 1024, 1147, 1247, 1254, 1260, 1313, 1358, 1463, 1469, 1538, 1615, 1728, 1919, 1962, 1970, 2047, 2071, 2090, 2204, 2299, 2321, 2345, 2404, 2625

Offset: 1

Labos Elemer, May 21 2003

nonn

			19 is a term since phi(19) = 18, phi(20) = 8, 8*18 = 144 = 12^2.

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

Cf. A000010, A083542, A066813, A058515, A083538, A083542, A083546.

f[x_] := EulerPhi[x]; Do[s=Sqrt[f[n+1]*f[n]]; If[IntegerQ[s], Print[n]], {n, 1, 5000}]
Position[Partition[EulerPhi[Range[2700]],2,1],?(IntegerQ[GeometricMean[ #]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, Sep 13 2020 *)

a(n) = x is such that sqrt(A000010(x)*A000010(x+1)) is an integer. Values of solutions x to phi(x) * phi(x+1) = A083542(x) = y^2.

A083550 Product of 2 consecutive prime differences of two successive terms of A001223.

Original entry on oeis.org

2, 4, 8, 8, 8, 8, 8, 24, 12, 12, 24, 8, 8, 24, 36, 12, 12, 24, 8, 12, 24, 24, 48, 32, 8, 8, 8, 8, 56, 56, 24, 12, 20, 20, 12, 36, 24, 24, 36, 12, 20, 20, 8, 8, 24, 144, 48, 8, 8, 24, 12, 20, 60, 36, 36, 12, 12, 24, 8, 20, 140, 56, 8, 8, 56, 84, 60, 20, 8, 24, 48, 48, 36, 24, 24, 48

Offset: 1

Labos Elemer, May 22 2003

nonn

Hauke Löffler, Table of n, a(n) for n = 1..10000

Cf. A001223, A083538-A083555, A057467.

f[x_] := Prime[x+1]-Prime[x]
Table[f[w+1]*f[w], {w, 1, 128}]

a(n) = A001223(n)*A001223(n+1) = (prime(n+1)-prime(n))*(prime(n+2)-prime(n+1)).

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 *)

A083551 Least common multiple of 2 consecutive prime differences, of two successive terms of A001223.

Original entry on oeis.org

2, 2, 4, 4, 4, 4, 4, 12, 6, 6, 12, 4, 4, 12, 6, 6, 6, 12, 4, 6, 12, 12, 24, 8, 4, 4, 4, 4, 28, 28, 12, 6, 10, 10, 6, 6, 12, 12, 6, 6, 10, 10, 4, 4, 12, 12, 12, 4, 4, 12, 6, 10, 30, 6, 6, 6, 6, 12, 4, 10, 70, 28, 4, 4, 28, 42, 30, 10, 4, 12, 24, 24, 6, 12, 12, 24, 8, 8, 40, 10, 10, 10, 6, 12

Offset: 1

Labos Elemer, May 22 2003

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A001223, A083538..A083555, A057467.

f[x_] := Prime[x+1]-Prime[x];  Table[LCM[f[w+1], f[w]], {w, 1, 128}]
Table[LCM[(Prime[n + 1] - Prime[n]), Prime[n + 2] - Prime[n + 1]], {n, 100}] (* Vincenzo Librandi, Mar 15 2018 *)
LCM@@#&/@Partition[Differences[Prime[Range[90]]],2,1] (* Harvey P. Dale, Oct 11 2020 *)

a(n) = lcm(A001223(n), A001223(n+1)).

A083554 Least common multiple of prime(n+1)-1 and prime(n)-1.

Original entry on oeis.org

2, 4, 12, 30, 60, 48, 144, 198, 308, 420, 180, 360, 840, 966, 1196, 1508, 1740, 660, 2310, 2520, 936, 3198, 3608, 1056, 2400, 5100, 5406, 5724, 3024, 1008, 8190, 8840, 9384, 10212, 11100, 3900, 4212, 13446, 14276, 15308, 16020, 3420, 18240, 9408

Offset: 1

Labos Elemer, May 22 2003

nonn

			n=25: a(25) = lcm(97-1, 101-1) = lcm(96,100) = 2400.

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

Cf. A006093, A083538-A083555, A058263.

f[x_] := Prime[x]-1; Table[LCM[f[w+1], f[w]], {w, 1, 128}]

a(n) = lcm(prime(n+1)-1, prime(n)-1); \\ Michel Marcus, Mar 15 2018

a(n) = lcm(A006093(n+1), A006093(n)) = lcm(prime(n+1)-1, prime(n)-1).

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))).

cp's OEIS Frontend

A083543 Duplicate of A083538.

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

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A083555 Quotient of LCM of prime(n+1)-1 and prime(n)-1 and GCD of the same two numbers.

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A083553 Product of prime(n+1)-1 and prime(n)-1.

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A083545 Numbers k such that the geometric mean of the Euler totient function of k and k+1 is an integer.

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A083550 Product of 2 consecutive prime differences of two successive terms of A001223.

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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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A083551 Least common multiple of 2 consecutive prime differences, of two successive terms of A001223.

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A083554 Least common multiple of prime(n+1)-1 and prime(n)-1.

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