A062354 -id:A062354 - cp's OEIS frontend

A009286 a(n) = lcm(sigma(n), phi(n)).

1, 3, 4, 14, 12, 12, 24, 60, 78, 36, 60, 28, 84, 24, 24, 248, 144, 78, 180, 168, 96, 180, 264, 120, 620, 84, 360, 168, 420, 72, 480, 1008, 240, 432, 48, 1092, 684, 180, 168, 720, 840, 96, 924, 420, 312, 792, 1104, 496, 798, 1860, 288, 1176, 1404, 360, 360, 120, 720, 1260

Offset: 1

David W. Wilson

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to lcm's

Cf. A000010, A000203, A009223, A062354, A084921.

A009286(n) = lcm(sigma(n), eulerphi(n)); \\ Antti Karttunen, May 26 2017

From Antti Karttunen, May 26 2017: (Start)
a(n) = A062354(n) / A009223(n).
a(A000040(n)) = A084921(n). - after Enrique Pérez Herrero's May 17 2012 comment in the latter sequence.
(End)

A065146 Numbers n such that the arithmetic, geometric and harmonic means of phi(n) and sigma(n) are all integers.

Original entry on oeis.org

1, 248, 264, 418, 477, 1485, 3080, 3135, 3596, 3828, 5396, 10098, 12648, 20026, 21318, 22152, 23374, 24882, 35074, 35343, 39105, 41656, 44660, 49938, 55154, 56536, 61344, 71145, 74613, 86304, 87087, 104931, 118296, 124605, 129504, 130356, 147560, 150195

Offset: 1

Labos Elemer, Oct 18 2001

nonn

			n = 5396, phi(n) = 2520, sigma(n) = 10080, arithmetic mean = 6300, geometric mean = 5040, harmonic mean = 4032; 67 cases < 10^6.

Harry J. Smith and Donovan Johnson, Table of n, a(n) for n = 1..5000 (first 500 terms from Harry J. Smith)

Cf. A000010, A000203, A062354, A011257.

Do[s = EulerPhi[n]*DivisorSigma[1, n]; z = (EulerPhi[n]+DivisorSigma[1, n])/2; u = h[n]; If[IntegerQ[Sqrt[s]]&&IntegerQ[z]&&IntegerQ[u], Print[n]], {n, 1, 1000000}]

{ n=0; for (m=1, 10^9, e=eulerphi(m); s=sigma(m); if (!issquare(e*s), next); h=(2*e*s)/(e + s); if (frac(h) != 0, next); if (frac((e + s)/2) != 0, next); write("b065146.txt", n++, " ", m); if (n==500, return) ) } \\ Harry J. Smith, Oct 12 2009

a = (phi(n)+sigma(n))/2, g = sqrt(phi(n)*sigma(n)), h = (2*phi(n)*sigma(n))/(phi(n)+sigma(n)) = g^2/a are all integers; phi() = A000010(), sigma() = A000203().

A065656 Composite numbers k such that sigma(k)phi(k) + 2(k+1) is a square.

Original entry on oeis.org

1169, 7777, 41111, 46097, 668167, 846817, 2107519, 3612769, 17424241, 30666527, 37526993, 56323393, 214746055, 383523857, 512376769, 1021934641, 1228492849, 1303949599, 4056001351, 7425397169, 17073544447, 17859428369, 18452226887, 46874737969, 51411954391

Offset: 1

Labos Elemer, Nov 12 2001

nonn

a(n) and square root of phi(a(n))*sigma(a(n)) + 2*a(n) + 2 are close to each other: e.g., a(7) = 2107519 and this square root is 2107458.
Since (p+1)*(p-1) + 2*(p+1) = p*p + 2*p + 1 = (p+1)^2 is a square, all primes are solutions.
73362272287 and 181264312447 are also terms. - Donovan Johnson, Jul 13 2012

			k = 7777: sigma(7777) = 9792, phi(7777) = 6000 and 9792*6000 + 2*7778 = 587675556 = 7666^2.

Cf. A062354, A000203, A000010, A065655.

isok(k) = { !isprime(k) && issquare(sigma(k)*eulerphi(k) + 2*(k + 1)) } \\ Harry J. Smith, Oct 26 2009

a(9)-a(15) from Harry J. Smith, Oct 26 2009
a(16)-a(20) from Donovan Johnson, May 24 2011
a(21)-a(25) from Donovan Johnson, Jul 13 2012

A056100 a(n) = sigma(n)*phi(n) + 1 (mod n).

Original entry on oeis.org

0, 0, 0, 3, 0, 1, 0, 5, 7, 3, 0, 5, 0, 5, 13, 9, 0, 1, 0, 17, 7, 9, 0, 1, 21, 11, 19, 1, 0, 7, 0, 17, 4, 15, 33, 13, 0, 17, 19, 1, 0, 19, 0, 9, 28, 21, 0, 17, 43, 11, 10, 13, 0, 1, 21, 25, 31, 27, 0, 49, 0, 29, 28, 33, 3, 43, 0, 21, 16, 27, 0, 1, 0, 35, 11, 25, 63, 55, 0, 33, 55, 39, 0, 1

Offset: 1

Robert G. Wilson v, Jul 28 2000

Note that iff p is a prime then sigma(p)*phi(p) + 1 = 0 (mod p).

George E. Andrews, "Number Theory," Dover Publ., NY, 1971, page 85.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A062354.

Table[Mod[DivisorSigma[1, n]*EulerPhi[n] + 1, n], {n, 1, 100}]

a(n) = (sigma(n)*eulerphi(n)+1) % n; \\ Michel Marcus, Aug 05 2025

from sympy import totient, divisor_sigma
def A056100(n): return (totient(n)*divisor_sigma(n)+1)%n # Karl-Heinz Hofmann, Aug 12 2025

A077101 a(n) = A051612(n)*A065387(n) = sigma(n)^2-phi(n)^2, where A051612(n) = sigma(n) - phi(n) and A065387(n) = sigma(n) + phi(n).

Original entry on oeis.org

0, 8, 12, 45, 20, 140, 28, 209, 133, 308, 44, 768, 52, 540, 512, 897, 68, 1485, 76, 1700, 880, 1196, 92, 3536, 561, 1620, 1276, 2992, 116, 5120, 124, 3713, 1904, 2660, 1728, 8137, 148, 3276, 2560, 7844, 164, 9072, 172, 6656, 5508, 4700, 188, 15120, 1485

Offset: 1

Labos Elemer, Nov 06 2002

If n is prime, then a(n) = 4n.

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

Cf. A000010, A000203, A002117, A051612, A062354, A065387, A065464, A072861, A077099, A077100, A127473.

Table[DivisorSigma[1,n]^2-EulerPhi[n]^2,{n,50}] (* Harvey P. Dale, Nov 08 2013 *)

A077101(n) = (sigma(n)^2 - eulerphi(n)^2); \\ Antti Karttunen, May 26 2017

a(n) = A077099(n) * A077100(n). - Antti Karttunen, May 26 2017
From Amiram Eldar, Dec 04 2023: (Start)
a(n) = A072861(n) - A127473(n).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 5*zeta(3)/2 - Product_{p prime}(1 - (2*p-1)/p^3) = (5/2)*A002117 - A065464 = 2.576892... . (End)

Edited by Dean Hickerson, Nov 07 2002

A093827 Decimal expansion of Silverman's constant.

Original entry on oeis.org

1, 7, 8, 6, 5, 7, 6, 4, 5, 9, 3, 6, 5, 9, 2, 2, 4, 6, 3, 4, 5, 8, 5, 9, 0, 4, 7, 5, 5, 4, 1, 3, 1, 5, 7, 5, 0, 3, 1, 2, 6, 2, 1, 9, 0, 2, 3, 8, 4, 2, 4, 3, 2, 9, 4, 9, 0, 1, 0, 7, 2, 4, 9, 6, 2, 1, 4, 2, 4, 5, 2, 7, 9, 1, 3, 4, 7, 8, 6, 2, 2, 3, 7, 7, 3, 2, 6, 9, 2, 4, 3, 9, 0, 3, 2, 8, 0, 5, 6, 8, 7, 6, 9, 0, 2

Offset: 1

Eric W. Weisstein, Apr 16 2004

Named after Robert D. Silverman. - Amiram Eldar, Aug 20 2020

			1.786576459365922463458590475541315750312621902384243294901...

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 161.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 3, p. 182.
Robert D. Silverman, A Peculiar Sum, USENET sci.math.research newsgroup posting, Mar 27 1996.

Steven R. Finch, Series Involving Arithmetic Functions, 2007.
Dave Rusin, Re: A peculiar sum, sci.math.research, Mar 27 1996.
Eric Weisstein's World of Mathematics, Silverman Constant.
Paul Zimmermann, Re: A Peculiar Sum USENET sci.math.research newsgroup posting, Mar 29 1996.

Cf. A000010, A000203, A062354.

read("transforms") ; Digits := 140 ; kmax := 450 ; tmax := kmax-10 ; 1+add(1/(p^(2*k)-p^(k-1)),k=1..kmax) : xt := subs(p=1/x,%) : xt := taylor(xt,x=0,tmax) ; L := [] ; for n from 1 to tmax-1 do L := [op(L),coeftayl(xt,x=0,n)]; end do: Le := EULERi(L) ; x := 1.0 ; for i from 2 to nops(Le) do x := x*Zeta(i)^op(i,Le) ; x := evalf(x) ; print(x) ; end do: # R. J. Mathar, Jul 28 2010

Sum[1/(EulerPhi[n]DivisorSigma[1, n]), {n, Infinity}]
$MaxExtraPrecision = 500; m = 500; f[p_] := 1 + Sum[1/(p^(2*k) - p^(k - 1)), {k, 1, 2*m}]; c = Rest@CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]*Range[m]; RealDigits[Exp[NSum[Indexed[c, n]*(PrimeZetaP[n])/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]] (* Amiram Eldar, Aug 20 2020 *)

From Amiram Eldar, Aug 20 2020: (Start)
Equals Sum_{k>=1} 1/(phi(k)*sigma(k)) = Sum_{k>=1} 1/A062354(k).
Equals Product_{p prime} (1 + Sum_{k>=1} 1/(p^(2*k) - p^(k-1))). (End)

37 more digits from R. J. Mathar, Jul 28 2010

More terms from Vaclav Kotesovec, Jun 13 2021

A130654 Exponent m such that 2^m = A092505(n) = A002430(n) / A046990(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1

Offset: 1

Alexander Adamchuk, Jun 20 2007, Jun 23 2007

nonn

Conjecture: A092505(n) is always a power of 2. a(n) = Log[ 2, A092505(n) ]. Note that a(n) = 0 iff n is a power of 2; or A002430(2^n) = A046990(2^n) and A092505(2^n) = 1. It appears that a(2k+1) = 1 for k>0. Note that least index k such that a(k) = n is {1, 3, 14, 60, ...} which apparently coincides with A006502(n) = {1, 3, 14, 60, 279, 1251, ...} Related to Fibonacci numbers (see Carlitz reference).
Least index k such that a(k) = n is listed in A131262(n) = {1, 3, 14, 60, 248, ...}. Conjecture: A131262(n) = Sigma(2^n)*EulerPhi(2^n) = 2^(2n) - Floor(2^n/2) = A062354(2^n). If this conjecture is true then a(1008) = 5 and a(n)<5 for all n<1008.
Positions of records indeed continue as 1, 3, 14, 60, 248, 1008, 4064, 16320, ..., strongly suggesting union of {1} and A171499. - Antti Karttunen, Jan 13 2019

			A092505(n) begins {1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 4, 2, 1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 2, 1, ...}.
Thus a(1) = Log[2,1] = 0, a(2) = Log[2,1] = 0, a(3) = Log[2,2] = 1.

Antti Karttunen, Table of n, a(n) for n = 1..16385
L. Carlitz, Some polynomials related to Fibonacci and Eulerian numbers, Fib. Quart., 16 (1978), 216-226.

Cf. A092505 = A002430(n) / A046990(n), n>0. Cf. A002430 = Numerators in Taylor series for tan(x). Cf. A046990 = Numerators of Taylor series for log(1/cos(x)). Cf. A006502 = Related to Fibonacci numbers.
Cf. A131262 = Least index k such that A130654(k) = n. Cf. A062354 = Sigma(n)*EulerPhi(n).
Cf. also A171499.

a=Series[ Tan[x], {x,0,256} ]; b=Series[ Log[ 1/Cos[x] ], {x,0,256}]; Table[ Log[ 2, Numerator[ SeriesCoefficient[ a, 2n-1 ] ] / Numerator[ SeriesCoefficient[ b, 2n ] ] ], {n,1,128} ]

a(n) = Log[ 2, A092505(n) ]. a(n) = Log[ 2, A002430(n) / A046990(n) ] = A007814(A092505(n)).

A015713 Numbers m such that phi(m) * sigma(m) + k^2 is not a square for any k.

Original entry on oeis.org

4, 9, 18, 49, 81, 98, 121, 162, 242, 361, 529, 722, 729, 961, 1058, 1458, 1849, 1922, 2209, 2401, 3481, 3698, 4418, 4489, 4802, 5041, 6241, 6561, 6889, 6962, 8978, 10082, 10609, 11449, 12482, 13122, 13778, 14641, 16129, 17161

Offset: 1

Robert G. Wilson v

nonn

Numbers m such that A062354(m) is in A016825. - Michel Marcus, Dec 07 2018

Amiram Eldar, Table of n, a(n) for n = 1..10000
Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.

Cf. A007814, A015710, A062354 (phi(n)*sigma(n)), A016825.

nonSqDiffQ[n_] := Mod[n, 4] == 2; aQ[n_] := nonSqDiffQ[ EulerPhi[n] * DivisorSigma[ 1, n]]; Select[Range[20000], aQ] (* Amiram Eldar, Dec 07 2018 *)

isok(n) = (sigma(n)*eulerphi(n) % 4) == 2; \\ Michel Marcus, Dec 07 2018

Conjecture: {4, p^(2*m), 2*p^(2*m), p = 4*k+3 is prime}. - Sean A. Irvine, Dec 06 2018

The conjecture is true. It can be proved using the multiplicative property of A062354(n), i.e., A062354(p^e) = p^(e-1)*(p^(e+1)-1), and that if m is a term then A007814(A062354(m)) = 1. - Amiram Eldar, Feb 11 2024

A065501 Number of conjugacy classes in the group SL(2,Z_n) (see A000056).

Original entry on oeis.org

1, 3, 7, 10, 9, 21, 11, 30, 25, 27, 15, 70, 17, 33, 63, 76, 21, 75, 23, 90, 77, 45, 27, 210, 49, 51, 79, 110, 33, 189, 35, 168, 105, 63, 99, 250, 41, 69, 119, 270, 45, 231, 47, 150, 225, 81, 51, 532, 81, 147, 147, 170, 57, 237, 135, 330, 161, 99, 63, 630, 65, 105, 275, 352, 153

Offset: 1

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 25 2001

Robin Visser, Table of n, a(n) for n = 1..500

Cf. A062354, A000056, A000252

[Nclasses(SpecialLinearGroup(2,ResidueClassRing(n))) : n in [2..50]]; // Robin Visser, Aug 06 2023

For an odd prime p : a(p) = p + 4. - Robin Visser, Aug 06 2023

a(23) corrected and more terms from Robin Visser, Aug 06 2023

A070732 Size of largest conjugacy class in the group GL(2,Z_n).

Original entry on oeis.org

1, 3, 12, 12, 30, 36, 56, 48, 108, 90, 132, 144, 182, 168, 360, 192, 306, 324, 380, 360, 672, 396, 552, 576, 750, 546, 972, 672, 870, 1080, 992, 768, 1584, 918, 1680, 1296, 1406, 1140, 2184, 1440, 1722, 2016, 1892, 1584, 3240, 1656, 2256, 2304, 2744, 2250

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 14 2002

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

Cf. A062354.
Cf. A000010, A000056.
Cf. A000082, A099892, A098198, A330523.

f[n_] := Block[{a = 1, b = FactorInteger[n]}, While[ Length[b] > 0, a = a*(b[[1, 1]] + 1)*b[[1, 1]]^(2b[[1, 2]] - If[ OddQ[ b[[1, 1]]], 1, 2]); b = Drop[b, 1]]; a]; Table[ f[n], {n, 1, 55}]
Table[n*Sum[d^2 MoebiusMu[n/d], {d, Divisors[n]}]/EulerPhi[2*n], {n, 1, 100}] (* Vaclav Kotesovec, Feb 01 2019 *)
f[p_, e_] := (p + 1)*p^(2*e - If[p == 2, 2, 1]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2020 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]+1)*f[i,1]^(2*f[i,2] - if(f[i,1]==2,2,1)));} \\ Amiram Eldar, Nov 05 2022

Multiplicative with a(p^e) = (p+1)*p^(2e - k), k = 1 if p is odd, k = 2 if p is 2.
a(n) = A000056(n)/A000010(2*n). - Vladeta Jovovic, Dec 22 2003
From R. J. Mathar, Apr 14 2011: (Start)
Dirichlet g.f.: (2^s-1)*zeta(s-1)*zeta(s-2)/((2^s+2)*zeta(2s-2)).
Dirichlet convolution of A000082 with a signed variant of A099892. (End)
Sum_{k=1..n} a(k) ~ 7*n^3 / (2*Pi^2). - Vaclav Kotesovec, Feb 01 2019
Sum_{n>=1} 1/a(n) = (13/11) * zeta(2)^2 * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = (13/11) * A098198 * A330523 = 1.7136743536... . - Amiram Eldar, Nov 05 2022

Edited by Robert G. Wilson v, May 20 2002

cp's OEIS Frontend

A009286 a(n) = lcm(sigma(n), phi(n)).

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A065146 Numbers n such that the arithmetic, geometric and harmonic means of phi(n) and sigma(n) are all integers.

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A065656 Composite numbers k such that sigma(k)*phi(k) + 2*(k+1) is a square.

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A056100 a(n) = sigma(n)*phi(n) + 1 (mod n).

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A077101 a(n) = A051612(n)*A065387(n) = sigma(n)^2-phi(n)^2, where A051612(n) = sigma(n) - phi(n) and A065387(n) = sigma(n) + phi(n).

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A093827 Decimal expansion of Silverman's constant.

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A130654 Exponent m such that 2^m = A092505(n) = A002430(n) / A046990(n).

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A065656 Composite numbers k such that sigma(k)phi(k) + 2(k+1) is a square.