A083538 -id:A083538 - cp's OEIS frontend

A083547 a(n) = sqrt(sqrt(phi(A083546(n)) * phi(1+A083546(n)))), the 4th root of product of totients of terms and 1+terms of A082788.

1, 12, 24, 36, 36, 60, 60, 72, 80, 96, 120, 120, 120, 144, 144, 168, 180, 240, 264, 360, 360, 432, 480, 504, 480, 480, 720, 720, 720, 720, 840, 840, 864, 840, 840, 840, 840, 960, 900, 960, 960, 1080, 1260, 1224, 1320, 1320, 1440, 1440, 1320, 1440, 1440, 1728

Offset: 1

Labos Elemer, May 21 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..300

Cf. A000010, A082788, A083538, A083542, A083545, A083546.

f[x_] := EulerPhi[x]; Do[s=Sqrt[Sqrt[f[n+1]*f[n]]]; If[IntegerQ[s], Print[s]], {n, 1, 1000000}]

a(29)-a(52) from Amiram Eldar, Apr 09 2021

A083548 Least common multiple of cototient values of consecutive integers.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 4, 12, 6, 6, 8, 8, 8, 56, 56, 8, 12, 12, 12, 36, 36, 12, 16, 80, 70, 126, 144, 16, 22, 22, 16, 208, 234, 198, 264, 24, 20, 60, 120, 24, 30, 30, 24, 168, 168, 24, 32, 224, 210, 570, 532, 28, 36, 180, 480, 672, 210, 30, 44, 44, 32, 864, 864, 544, 782, 46, 36

Offset: 1

Labos Elemer, May 22 2003

nonn

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

Cf. A051953, A083538-A083555, A049586.

f[x_] := x-EulerPhi[x] Table[LCM[f[w+1], f[w]], {w, 1, b}]
LCM@@#&/@Partition[Table[n-EulerPhi[n],{n,70}],2,1] (* Harvey P. Dale, Feb 01 2015 *)

a(n)=lcm(n-eulerphi(n),n+1-eulerphi(n+1)) \\ Charles R Greathouse IV, Nov 16 2012

a(n) = lcm(A051953(n), A051953(n+1)).

A083552 Quotient when LCM of 2 consecutive prime differences is divided by GCD of the same two differences.

Original entry on oeis.org

2, 1, 2, 2, 2, 2, 2, 6, 3, 3, 6, 2, 2, 6, 1, 3, 3, 6, 2, 3, 6, 6, 12, 2, 2, 2, 2, 2, 14, 14, 6, 3, 5, 5, 3, 1, 6, 6, 1, 3, 5, 5, 2, 2, 6, 1, 3, 2, 2, 6, 3, 5, 15, 1, 1, 3, 3, 6, 2, 5, 35, 14, 2, 2, 14, 21, 15, 5, 2, 6, 12, 12, 1, 6, 6, 12, 2, 2, 20, 5, 5, 5, 3, 6, 6, 12, 2, 2, 2, 3, 6, 2, 2, 2, 6, 2, 6, 9

Offset: 1

Labos Elemer, May 22 2003

nonn

Conjecture: Every positive integer appears infinitely many times in this sequence. Example: a(834) = a(909) = ... = a(9901) = ... = 4. - Jerzy R Borysowicz, Dec 22 2018

All terms of this sequence are integers because gcd(r,s) divides lcm(r,s) for any r and s. - Jerzy R Borysowicz, Jan 05 2019

Amiram Eldar, 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]]/GCD[f[w+1], f[w]], {w, 1, 128}]

a(n) = my(da=prime(n+2)-prime(n+1), db=prime(n+1)-prime(n)); lcm(da, db)/gcd(da, db) \\ Felix Fröhlich, Jan 05 2019

a(n) = lcm(A001223(n), A001223(n+1))/gcd(A001223(n), A001223(n+1));

a(n) = A083551(n)/A057467(n).

A082849 Product of cototient values of consecutive integers.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 4, 12, 18, 6, 8, 8, 8, 56, 56, 8, 12, 12, 12, 108, 108, 12, 16, 80, 70, 126, 144, 16, 22, 22, 16, 208, 234, 198, 264, 24, 20, 300, 360, 24, 30, 30, 24, 504, 504, 24, 32, 224, 210, 570, 532, 28, 36, 540, 480, 672, 630, 30, 44, 44, 32, 864, 864, 544, 782

Offset: 1

Labos Elemer, May 22 2003

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Laurenţiu Panaitopol, Asymptotical formula for a(n) = n - e(n), Bull. Math. Soc. Sci. Math. Roumanie, Vol. 42 (90), No. 3 (1999), pp. 271-277.

Cf. A051953, A065474, A083538-A083555.

f[x_] := x-EulerPhi[x] tpr=Table[f[w+1]*f[w], {w, 1, 128}]
Times@@@Partition[Table[n-EulerPhi[n],{n,70}],2,1] (* Harvey P. Dale, Nov 17 2020 *)

p=2;forprime(q=3,97,print1((p-eulerphi(p))*(q-eulerphi(q))", ");p=q) \\ Charles R Greathouse IV, Nov 16 2012

a(n) = A051953(n) * A051953(n+1).

Sum_{k=1..n} a(k) ~ c * n^3 + O(n^2 * log(n)^2), where c = (1/3) * (1 + Product_{p prime} (1 - 2/p^2)) - 4/Pi^2 = 0.03559329841.... (Panaitopol, 1999). - Amiram Eldar, Mar 09 2021

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

A083547 a(n) = sqrt(sqrt(phi(A083546(n)) * phi(1+A083546(n)))), the 4th root of product of totients of terms and 1+terms of A082788.

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A083548 Least common multiple of cototient values of consecutive integers.

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A083552 Quotient when LCM of 2 consecutive prime differences is divided by GCD of the same two differences.

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A082849 Product of cototient values of consecutive integers.

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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.

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