A083542 -id:A083542 - cp's OEIS frontend

A238453 Triangle read by rows: T(n,k) = A001088(n)/(A001088(k)*A001088(n-k)).

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 4, 8, 8, 4, 1, 1, 2, 8, 8, 8, 2, 1, 1, 6, 12, 24, 24, 12, 6, 1, 1, 4, 24, 24, 48, 24, 24, 4, 1, 1, 6, 24, 72, 72, 72, 72, 24, 6, 1, 1, 4, 24, 48, 144, 72, 144, 48, 24, 4, 1, 1, 10, 40, 120, 240, 360, 360, 240, 120

Offset: 0

Tom Edgar, Feb 26 2014

We assume that A001088(0)=1 since it would be the empty product.
These are the generalized binomial coefficients associated with Euler's totient function A000010.
Another name might be the totienomial coefficients.

			The first five terms in Euler's totient function are 1,1,2,2,4 and so T(4,2) = 2*2*1*1/((1*1)*(1*1))=4 and T(5,3) = 4*2*2*1*1/((2*1*1)*(1*1))=8.
The triangle begins
1
1 1
1 1 1
1 2 2 1
1 2 4 2 1
1 4 8 8 4 1
1 2 8 8 8 2 1

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Tom Edgar, Totienomial Coefficients, INTEGERS, 14 (2014), #A62.
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.
Donald E. Knuth and Herbert S. Wilf, The power of a prime that divides a generalized binomial coefficient, J. Reine Angew. Math., 396:212-219, 1989.

Cf. A000010, A001088, A083542.

a238453 n k = a238453_tabl !! n !! k
a238453_row n = a238453_tabl !! n
a238453_tabl = [1] : f [1] a000010_list where
   f xs (z:zs) = (map (div y) $ zipWith (*) ys $ reverse ys) : f ys zs
     where ys = y : xs; y = head xs * z
-- Reinhard Zumkeller, Feb 27 2014

f[n_] := Product[EulerPhi@ k, {k, n}]; Table[f[n]/(f[k] f[n - k]), {n, 0, 11}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 19 2016 *)

T(n,k)={prod(i=1, k, eulerphi(n+1-i)/eulerphi(i))} \\ Andrew Howroyd, Nov 13 2018

q=100 #change q for more rows
P=[euler_phi(i) for i in [0..q]]
[[prod(P[1:n+1])/(prod(P[1:k+1])*prod(P[1:(n-k)+1])) for k in [0..n]] for n in [0..len(P)-1]] #generates the triangle up to q rows.

T(n,k) = A001088(n)/(A001088(k)*A001088(n-k)).
T(n,k) = prod_{i=1..n} A000010(i)/(prod_{i=1..k} A000010(i)*prod_{i=1..n-k} A000010(i)).
T(n,k) = A000010(n)/n*(k/A000010(k)*T(n-1,k-1)+(n-k)/A000010(n-k)*T(n-1,k)).
T(n+1, 2) = A083542(n). - Michael Somos, Aug 26 2014
T(n,k) = Product_{i=1..k} (phi(n+1-i)/phi(i)), where phi is Euler's totient function (A000010). - Werner Schulte, Nov 14 2018

A058515 GCD of totients of consecutive integers.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 6, 2, 8, 8, 2, 6, 2, 4, 2, 2, 2, 4, 4, 6, 6, 4, 4, 2, 2, 4, 4, 8, 12, 12, 18, 6, 8, 8, 4, 6, 2, 4, 2, 2, 2, 2, 2, 4, 8, 4, 2, 2, 8, 12, 4, 2, 2, 4, 30, 6, 4, 16, 4, 2, 2, 4, 4, 2, 2, 24, 36, 4, 4, 12, 12, 6, 2, 2, 2, 2, 2, 8, 2, 14, 8, 8, 8, 24, 4, 4, 2, 2, 8, 32, 6

Offset: 1

Labos Elemer, Dec 21 2000

			For n = 61, gcd(phi(62), phi(61)) = gcd(30, 60) = 30, so a(61) = 30.

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

Cf. A000010, A049586, A060778, A066813, A083542.

Map[GCD @@ # &, Partition[EulerPhi@ Range@ 98, 2, 1]] (* Michael De Vlieger, Aug 22 2017 *)

a(n) = gcd(eulerphi(n), eulerphi(n+1)); \\ Michel Marcus, Dec 10 2013

a(n) = gcd(phi(n+1), phi(n)), where phi = A000010.

a(n) = A083542(n)/A066813(n). - Amiram Eldar, May 07 2025

Offset corrected to 1 by Michel Marcus, Dec 10 2013

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

A066813 a(n) = lcm(phi(n), phi(n+1)).

Original entry on oeis.org

1, 2, 2, 4, 4, 6, 12, 12, 12, 20, 20, 12, 12, 24, 8, 16, 48, 18, 72, 24, 60, 110, 88, 40, 60, 36, 36, 84, 56, 120, 240, 80, 80, 48, 24, 36, 36, 72, 48, 80, 120, 84, 420, 120, 264, 506, 368, 336, 420, 160, 96, 312, 468, 360, 120, 72, 252, 812, 464, 240, 60, 180, 288, 96

Offset: 1

Benoit Cloitre, Jan 20 2002

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000010 (phi), A058515, A083542.

LCM@@EulerPhi[#]&/@Partition[Range[100],2,1] (* Harvey P. Dale, May 07 2011 *)

a(n) = { lcm(eulerphi(n), eulerphi(n+1)) } \\ Harry J. Smith, Mar 29 2010

a(n) = A083542(n)/A058515(n). - Amiram Eldar, May 07 2025

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.

A083546 The geometric mean of the Euler totient function of 2 consecutive integers {k, k+1} when it is an integer.

Original entry on oeis.org

1, 2, 8, 12, 48, 48, 60, 80, 96, 128, 144, 180, 280, 240, 240, 288, 288, 288, 336, 288, 384, 360, 480, 480, 640, 720, 672, 600, 576, 720, 720, 720, 672, 864, 960, 864, 960, 1080, 1008, 1408, 1296, 960, 1008, 1320, 1260, 1056, 1440, 1200, 1728, 1440, 1296

Offset: 1

Labos Elemer, May 21 2003

nonn

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

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

Cf. A000010, A083542, A083542, A066813, A058515, A083542, A083545.

f[x_] := EulerPhi[x]; Do[s=Sqrt[f[n+1]*f[n]]; If[IntegerQ[s], Print[s]], {n, 1, 5000}]
Select[Sqrt[#]&/@(Times@@@Partition[EulerPhi[Range[3000]],2,1]),IntegerQ] (* Harvey P. Dale, Nov 04 2020 *)

a(n) = sqrt(A000010(x) * A000010(x+1)) = sqrt(phi(x) * phi(x+1)) = sqrt(A083542(x)) where x = A083545(n).

A330319 a(n) = Sum_{i=1..n} phi(i)*phi(i+1), where phi(n) = A000010(n) is Euler's totient function.

Original entry on oeis.org

1, 3, 7, 15, 23, 35, 59, 83, 107, 147, 187, 235, 307, 355, 419, 547, 643, 751, 895, 991, 1111, 1331, 1507, 1667, 1907, 2123, 2339, 2675, 2899, 3139, 3619, 3939, 4259, 4643, 4931, 5363, 6011, 6443, 6827, 7467, 7947, 8451, 9291, 9771, 10299, 11311, 12047, 12719, 13559, 14199, 14967, 16215, 17151, 17871, 18831

Offset: 1

N. J. A. Sloane, Dec 11 2019

nonn

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 32.

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.
L. Mirsky, Summation formula involving arithmetic functions, Duke Mathematical Journal, Vol. 16, No. 2 (1949), pp. 261-272.

Cf. A000010, A065474, A335131.

Partial sums of A083542.

phi = EulerPhi[Range[56]]; Accumulate[Most[phi] * Rest[phi]] (* Amiram Eldar, Mar 05 2020 *)

a(n) = sum(i=1, n, eulerphi(i)*eulerphi(i+1)); \\ Michel Marcus, Mar 05 2020

a(n) ~ (c/3) * n^3 + O(n^2*log(n)^2), where c = Product_{p prime}(1 - 2/p^2) (A065474). - Amiram Eldar, Mar 05 2020

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.

Original entry on oeis.org

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

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

A238453 Triangle read by rows: T(n,k) = A001088(n)/(A001088(k)*A001088(n-k)).

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A058515 GCD of totients of consecutive integers.

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

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A066813 a(n) = lcm(phi(n), phi(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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A083546 The geometric mean of the Euler totient function of 2 consecutive integers {k, k+1} when it is an integer.

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A330319 a(n) = Sum_{i=1..n} phi(i)*phi(i+1), where phi(n) = A000010(n) is Euler's totient function.

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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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