A062354 -id:A062354 - cp's OEIS frontend

A306724 Least number k > 1 such that A062354(k) is an n-th power, where A062354 is the product of sigma (A000203) and phi (A000010).

2, 14, 3, 170, 3570, 592922491, 17194752239, 498892319051, 14467877252479, 421652049419104, 12227909433154016, 377536703748630244, 926952707565364023467, 1485824943636552705389704010031591742370238670767627108613, 18031470774665264926975299618474551942701594055200456829621877, 219559123426400144842876467461078524942414020727022446946702813568

Offset: 1

Amiram Eldar, Mar 06 2019

nonn

10^70 < a(17) <= 842075173279428103504117746722346581987825143261978794674517195307859044272000. - Max Alekseyev, Oct 07 2023

			A062354(14) = 12^2;
A062354(3) = 2^3;
A062354(170) = 12^4;
A062354(3570) = 24^5;
A062354(592922491) = 840^6;
A062354(17194752239) = 840^7.

Cf. A000010, A000203, A011257, A062354, A114077, A114078.

a[n_] := Module[{k=2}, While[!IntegerQ[Surd[DivisorSigma[1, k]*EulerPhi[k], n]], k++]; k]; Array[a, 1, 5]

a(n) = {my(k=2); while (!ispower(sigma(k)*eulerphi(k), n), k++); k;} \\ Michel Marcus, Mar 06 2019

a(8) from Giovanni Resta, Mar 06 2019

a(9)-a(13) from Daniel Suteu confirmed, a(14)-a(16) added by Max Alekseyev, Oct 06 2023

A014574 Average of twin prime pairs.

Original entry on oeis.org

4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, 312, 348, 420, 432, 462, 522, 570, 600, 618, 642, 660, 810, 822, 828, 858, 882, 1020, 1032, 1050, 1062, 1092, 1152, 1230, 1278, 1290, 1302, 1320, 1428, 1452, 1482, 1488, 1608

Offset: 1

R. K. Guy, N. J. A. Sloane, Eric W. Weisstein

With an initial 1 added, this is the complement of the closure of {2} under a*b+1 and a*b-1. - Franklin T. Adams-Watters, Jan 11 2006
Also the square root of the product of twin prime pairs + 1. Two consecutive odd numbers can be written as 2k+1,2k+3. Then (2k+1)(2k+3)+1 = 4(k^2+2k+1) = 4(k+1)^2, a perfect square. Since twin prime pairs are two consecutive odd numbers, the statement is true for all twin prime pairs. - Cino Hilliard, May 03 2006
Or, single (or isolated) composites. Nonprimes k such that neither k-1 nor k+1 is nonprime. - Juri-Stepan Gerasimov, Aug 11 2009
Numbers n such that sigma(n-1) = phi(n+1). - Farideh Firoozbakht, Jul 04 2010
Aside from the first term in the sequence, all remaining terms have digital root 3, 6, or 9. - J. W. Helkenberg, Jul 24 2013
Numbers n such that n^2-1 is a semiprime. - Thomas Ordowski, Sep 24 2015
Every term but the first is a multiple of 6. - Harvey P. Dale, Mar 31 2023

Archimedeans Problems Drive, Eureka, 30 (1967).

T. D. Noe, Table of n, a(n) for n = 1..10000
C. K. Caldwell, The Prime Glossary: Twin primes
C. K. Caldwell, The Top Twenty: Twin Primes
Y. Fujiwara, Parsing a Sequence of Qubits, IEEE Trans. Information Theory, 59 (2013), 6796-6806.
Y. Fujiwara, Parsing a Sequence of Qubits, arXiv:1207.1138 [quant-ph], 2012-2013.
L. J. Gerstein, A reformulation of the Goldbach conjecture, Math. Mag., 66 (1993), 44-45.
Brian Hayes, Does having prime neighbors make you more composite?, Bit-Player Article, Nov 04 2021
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A000010, A000203, A001359, A002822, A006512, A037074, A040040, A054735, A077800, A111046.

A068507 is the intersection of A002182 and this sequence.

a:=1+Filtered([1..2000],p->IsPrime(p) and IsPrime(p+2)); # Muniru A Asiru, May 20 2018

a014574 n = a014574_list !! (n-1)
a014574_list = [x | x <- [2,4..], a010051 (x-1) == 1, a010051 (x+1) == 1]
-- Reinhard Zumkeller, Apr 11 2012

P := select(isprime,[$1..1609]): map(p->p+1,select(p->member(p+2,P),P)); # Peter Luschny, Mar 03 2011
A014574 := proc(n) option remember; local p ; if n = 1 then 4 ; else p := nextprime( procname(n-1) ) ; while not isprime(p+2) do p := nextprime(p) ; od ; return p+1 ; end if ; end proc: # R. J. Mathar, Jun 11 2011

Select[Table[Prime[n] + 1, {n, 260}], PrimeQ[ # + 1] &] (* Ray Chandler, Oct 12 2005 *)
Mean/@Select[Partition[Prime[Range[300]],2,1],Last[#]-First[#]==2&] (* Harvey P. Dale, Jan 16 2014 *)

A014574(n) := block(
    if n = 1 then
        return(4),
    p : A014574(n-1) ,
    for k : 2 step 2 do (
        if primep(p+k-1) and primep(p+k+1) then
            return(p+k)
    )
)$ /* R. J. Mathar, Mar 15 2012 */

p=2;forprime(q=3,1e4,if(q-p==2,print1(p+1", "));p=q) \\ Charles R Greathouse IV, Jun 10 2011

a(n) = (A001359(n) + A006512(n))/2 = 2*A040040(n) = A054735(n)/2 = A111046(n)/4.
a(n) = A129297(n+4). - Reinhard Zumkeller, Apr 09 2007
A010051(a(n) - 1) * A010051(a(n) + 1) = 1. Reinhard Zumkeller, Apr 11 2012
a(n) = 6*A002822(n-1), n>=2. - Ivan N. Ianakiev, Aug 19 2013
a(n)^4 - 4*a(n)^2 = A062354(a(n)^2 - 1). - Raphie Frank, Oct 17 2013

Offset changed to 1 by R. J. Mathar, Jun 11 2011

A065465 Decimal expansion of Product_{p prime} (1 - 1/(p^2*(p+1))).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 19 2001

From Richard R. Forberg, May 22 2023: (Start)
This constant is the asymptotic mean of (phi(n)/n)*(sigma(n)/n), where phi is the Euler totient function (A000010) and sigma is the sum-of-divisors function (A000203).
In contrast, the product of the separate means, mean(phi(n)/n) * mean(sigma(n)/n), converges to 1, with the asymptotic mean(sigma(n)/n) = Pi^2/6 = zeta(2). See A013661.
Also see A062354. (End)

			0.88151383972517077692839182290...

Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Zeitschr., Vol. 74 (1960), pp. 66-80.
Steven R. Finch, Class number theory, page 7. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 50 and 85.
R. J. Mathar, Hardy-Littlewood Constants Embedded into Infinite Products over All Positive Integers, arXiv:0903.2514 [math.NT], 2009-2011, Table 5, constant Q_1^(2).
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
Simon Plouffe, Generalized expansions of real numbers, 2006.
Eric Weisstein's World of Mathematics, Quadratic Class Number Constant.
Eric Weisstein's World of Mathematics, Prime Products.

Cf. A000010, A007434, A013661, A062354, A078087.

$MaxExtraPrecision = 1000; digits = 98; terms = 1000; LR = Join[{0, 0, 0}, LinearRecurrence[{-2, -1, 1, 1}, {-3, 4, -5, 3}, terms+10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*PrimeZetaP[n-1]/(n-1), {n, 4, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 16 2016 *)

prodeulerrat(1 - 1/(p^2*(p+1))) \\ Amiram Eldar, Mar 14 2021

Sum_{n>=1} phi(n)/(n*J(n)) = (this constant)*A013661 with phi()=A000010() and J() = A007434() [Cohen, Corollary 5.1.1]. - R. J. Mathar, Apr 11 2011

A062355 a(n) = d(n) * phi(n), where d(n) is the number of divisors function.

Original entry on oeis.org

1, 2, 4, 6, 8, 8, 12, 16, 18, 16, 20, 24, 24, 24, 32, 40, 32, 36, 36, 48, 48, 40, 44, 64, 60, 48, 72, 72, 56, 64, 60, 96, 80, 64, 96, 108, 72, 72, 96, 128, 80, 96, 84, 120, 144, 88, 92, 160, 126, 120, 128, 144, 104, 144, 160, 192, 144, 112, 116, 192, 120, 120, 216, 224

Offset: 1

Jason Earls, Jul 06 2001

a(n) = sum of gcd(k-1,n) for 1 <= k <= n and gcd(k,n)=1 (Menon's identity).
For n = 2^(4*k^2 - 1), k >= 1, the terms of the sequence are square and for n = 2^((3*k + 2)^3 - 1), k >= 1, the terms of the sequence are cubes. - Marius A. Burtea, Nov 14 2019
Sum_{k>=1} 1/a(k) diverges. - Vaclav Kotesovec, Sep 20 2020

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, Prob. 7.2 12, p. 141.
P. K. Menon, On the sum gcd(a-1,n) [(a,n)=1], J. Indian Math. Soc. (N.S.), 29 (1965), 155-163.
József Sándor, On Dedekind's arithmetical function, Seminarul de teoria structurilor (in Romanian), No. 51, Univ. Timișoara, 1988, pp. 1-15. See p. 11.
József Sándor, Some diophantine equations for particular arithmetic functions (in Romanian), Seminarul de teoria structurilor, No. 53, Univ. Timișoara, 1989, pp. 1-10. See p. 8.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..1000 from Harry J. Smith)
Pentti Haukkanen and László Tóth, Menon-type identities again: Note on a paper by Li, Kim and Qiao, arXiv:1911.05411 [math.NT], 2019.
Vaclav Kotesovec, Graph - the asymptotic ratio (250000000 terms)
R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions, arXiv:1106.4038 [math.NT], 2011-2012, Section 3.15.
R. Sivaramakrishnan, Problem E 1962, Elementary Problems, The American Mathematical Monthly, Vol. 74, No. 2 (1967), p. 198; Solution, ibid., Vol. 75, No. 5 (1968), p. 550.
Marius Tarnauceanu, A generalization of the Menon's identity, arXiv:1109.2198 [math.GR], 2011-2012.
Laszlo Toth, Menon's identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110.
Wikipedia, Arithmetic function (Menon's identity).

Cf. A003557, A173557, A061468, A062816, A079535, A062949 (inverse Mobius transform), A304408, A318519, A327169 (number of times n occurs in this sequence).

Cf. A062354, A064840.

[NumberOfDivisors(n)*EulerPhi(n):n in [1..65]]; // Marius A. Burtea, Nov 14 2019

seq(tau(n)*phi(n), n=1..64); # Zerinvary Lajos, Jan 22 2007

Table[EulerPhi[n] DivisorSigma[0, n], {n, 80}] (* Carl Najafi, Aug 16 2011 *)
f[p_, e_] := (e+1)*(p-1)*p^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 21 2020 *)

a(n)=numdiv(n)*eulerphi(n); vector(150,n,a(n))

{ for (n=1, 1000, write("b062355.txt", n, " ", numdiv(n)*eulerphi(n)) ) } \\ Harry J. Smith, Aug 05 2009

for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X + p*X^2)/(1 - p*X)^2)[n], ", ")) \\ Vaclav Kotesovec, Jun 15 2020

Dirichlet convolution of A047994 and A000010. - R. J. Mathar, Apr 15 2011
a(n) = A000005(n)*A000010(n). Multiplicative with a(p^e) = (e+1)*(p-1)*p^(e-1). - R. J. Mathar, Jun 23 2018
a(n) = A173557(n) * A318519(n) = A003557(n) * A304408(n). - Antti Karttunen, Sep 16 2018 & Sep 20 2019
From Vaclav Kotesovec, Jun 15 2020: (Start)
Let f(s) = Product_{primes p} (1 - 2*p^(-s) + p^(1-2*s)).
Dirichlet g.f.: zeta(s-1)^2 * f(s).
Sum_{k=1..n} a(k) ~ n^2 * (f(2)*(log(n)/2 + gamma - 1/4) + f'(2)/2), where f(2) = A065464 = Product_{primes p} (1 - 2/p^2 + 1/p^3) = 0.42824950567709444...,
f'(2) = 2 * A065464 * A335707 = f(2) * Sum_{primes p} 2*log(p) / (p^2 + p - 1) = 0.35866545223424232469545420783620795... and gamma is the Euler-Mascheroni constant A001620. (End)
From Amiram Eldar, Mar 02 2021: (Start)
a(n) >= n (Sivaramakrishnan, 1967).
a(n) >= sigma(n), for odd n (Sándor, 1988).
a(n) >= phi(n) + n - 1 (Sándor, 1989) (End)
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} uphi(gcd(n,k)), where uphi(n) = A047994(n).
a(n) = Sum_{k=1..n} uphi(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

A064840 a(n) = tau(n)*sigma(n).

Original entry on oeis.org

1, 6, 8, 21, 12, 48, 16, 60, 39, 72, 24, 168, 28, 96, 96, 155, 36, 234, 40, 252, 128, 144, 48, 480, 93, 168, 160, 336, 60, 576, 64, 378, 192, 216, 192, 819, 76, 240, 224, 720, 84, 768, 88, 504, 468, 288, 96, 1240, 171, 558, 288, 588, 108, 960, 288, 960, 320, 360

Offset: 1

Vladeta Jovovic, Oct 25 2001

Dirichlet convolution of A034761 with (the Dirichlet inverse of A037213). - R. J. Mathar, Feb 11 2011

			For n = 10, a(10) = sigma(10) * tau(10) = 18 * 4 = 72. - _Indranil Ghosh_, Jan 20 2017

Indranil Ghosh, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
Vaclav Kotesovec, Graph - the asymptotic ratio

Cf. A000005, A000203, A007503, A062354, A062355.

[ NumberOfDivisors(n)*SumOfDivisors(n) : n in [1..40]];

with(numtheory): seq(sigma(n)*tau(n), n=1..58) ; # Zerinvary Lajos, Jun 04 2008

Table[ DivisorSigma[0, n] * DivisorSigma[1, n], {n, 1, 58}] (* Jean-François Alcover, Mar 26 2013 *)

{ for (n=1, 1000, a=numdiv(n)*sigma(n); write("b064840.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 28 2009

Multiplicative with a(p^e) = (p^(e+1)-1)*(e+1)/(p-1). a(n) = (1/2)*Sum_{i|n, j|n} (i+j).
Dirichlet g.f. (zeta(s)*zeta(s-1))^2/zeta(2s-1). - R. J. Mathar, Feb 11 2011
Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / (144*Zeta(3)) * (2*log(n) - 1 + 4*gamma - 4*Zeta'(3)/Zeta(3) + 24*Zeta'(2)/Pi^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 31 2019

A330523 Decimal expansion of Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Dec 17 2019

			0.5358961538283379998085026313185459506482223745141452711510108346133288119...

Cf. A055653, A062354, A065465, A175639, A256392, A332713.

Do[Print[N[Exp[-Sum[q = Expand[(p^2 + p^3 - p^4)^j]; Sum[PrimeZetaP[Exponent[q[[k]], p]] * Coefficient[q[[k]], p^Exponent[q[[k]], p]], {k, 1, Length[q]}]/j, {j, 1, t}]], 50]], {t, 10, 100, 10}]

(6/Pi^2) * prodeulerrat(1 - 1/(p^2*(p+1))) \\ Amiram Eldar, Sep 08 2020

Equals (6/Pi^2) * A065465. - Amiram Eldar, Sep 08 2020

A062952 Multiplicative with a(p^e) = (p^(2*e+2)-p^(e+1)-p^e+p)/(p^2-1).

Original entry on oeis.org

1, 4, 9, 18, 25, 36, 49, 78, 87, 100, 121, 162, 169, 196, 225, 326, 289, 348, 361, 450, 441, 484, 529, 702, 645, 676, 807, 882, 841, 900, 961, 1334, 1089, 1156, 1225, 1566, 1369, 1444, 1521, 1950, 1681, 1764, 1849, 2178, 2175, 2116, 2209, 2934, 2443, 2580

Offset: 1

Vladeta Jovovic, Jul 21 2001

If k is squarefree (cf. A005117) then A062952(k) = k^2. - Benoit Cloitre, Apr 16 2002

Inverse Möbius transform of A062354(n). - Wesley Ivan Hurt, Jul 26 2025

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

Cf. A005117, A029939, A057660, A062952.
Cf. A002117, A013661, A183699, A330523.
Cf. A062354.

f[p_, e_] := (p^(2*e+2)-p^(e+1)-p^e+p)/(p^2-1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 50] (* Amiram Eldar, Jul 31 2019 *)

a(n) = sumdiv(n, d, eulerphi(d)*sigma(d)) \\ Michel Marcus, Jun 17 2013

a(n) = Sum_{d|n} phi(d)*sigma(d).
a(n) = Sum_{k=1..n} sigma(n/gcd(n, k)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(2)*zeta(3)/3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A183699 * A330523 / 3. - Amiram Eldar, Oct 30 2022

A011775 Numbers k such that k divides phi(k) * sigma(k).

Original entry on oeis.org

1, 6, 18, 24, 28, 40, 54, 72, 84, 96, 117, 120, 135, 162, 196, 200, 216, 224, 234, 252, 270, 288, 360, 384, 468, 486, 496, 540, 588, 600, 640, 648, 672, 756, 775, 819, 864, 891, 936, 1000, 1080, 1152, 1350, 1372, 1458, 1488, 1521, 1536, 1550, 1568, 1638, 1701, 1764

Offset: 1

R. K. Guy

nonn

Comments from Farideh Firoozbakht, Dec 01 2005: (Start)
I. All numbers of the form 2^(4m-1)*5^n where m & n are natural numbers are in the sequence. Because if s=2^(4m-1)*5^n then phi(s)=2^(4m-2)*4*5^(n-1); sigma(s)=(2^(4m)-1)*(5^(n+1)-1)/4 so phi(s)*sigma(s)=6*((16^m-1)/15)*((5^(n+1)-1)/4)*(2^(4m-1)*5^n)= 6*((16^m-1)/15)*((5^(n+1)-1)/4)*s, note that (16^m-1)/15 and (5^(n+1)-1)/4 are integers, hence s divides phi(s)*sigma(s).
II. All numbers of the form 2^(2m-1)*3^n where m & n are natural numbers (A228104) are in the sequence. Because if s=2^(2m-1)*3^n then phi(s)=2^(2m-2)*2*3^(n-1); sigma(s)=(2^(2m)-1)*(3^(n+1)-1)/2 so phi(s)*sigma(s)=((3^(n+1)-1)/2)*((4^m-1)/3)*(2^(2m-1)*3^n) =((3^(n+1)-1)/2)*((4^m-1)/3)*s, note that (3^(n+1)-1)/2 and (4^m-1)/3 are integers, hence s divides phi(s)*sigma(s).
So this sequence is infinite. Also it is obvious that perfect numbers (A000396) and multiply-perfect numbers(A007691) are subsequences of this sequence. (End)

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

Cf. A062354, A000396, A007691.

Select[Range[1770], IntegerQ[DivisorSigma[1, # ]*EulerPhi[ # ]/# ] &] (* Farideh Firoozbakht, Dec 01 2005 *)

is(n)=sigma(n)*eulerphi(n)%n==0 \\ Charles R Greathouse IV, Nov 27 2013

Corrected and extended by David W. Wilson

A065299 Numbers k such that sigma(k)*phi(k) is squarefree.

Original entry on oeis.org

1, 2, 4, 9, 121, 242, 529, 1058, 2209, 3481, 4418, 5041, 6889, 6962, 10082, 11449, 13778, 17161, 22898, 27889, 32041, 34322, 51529, 55778, 57121, 64082, 96721, 103058, 114242, 120409, 128881, 146689, 175561, 185761, 193442, 196249, 218089

Offset: 1

Labos Elemer, Oct 29 2001

nonn

			All solutions are either squares or twice squares. Proper subset of A055008 or A028982. Several squares (of primes) and 2*squares are not here. E.g., 242 is here because phi(242) = 110, sigma(242) = 399, 2*5*11*3*7*19 is squarefree; 18 is not here, since 2*3*3*13 is not squarefree.

Harry J. Smith and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (terms 1..500 from Smith)

A062354(m) is squarefree.

Cf. A000010, A000203, A008683, A055008, A028982, A049195-A049199, A049149.

a[x_] := Abs[MoebiusMu[DivisorSigma[1, x]*EulerPhi[x]]] Do[s=as[n]; If[Equal[s, 1], Print[{n, Sqrt[n]}]], {n, 1, 1000000}]
Select[Range[250000],SquareFreeQ[DivisorSigma[1,#]*EulerPhi[#]]&] (* Harvey P. Dale, Jul 15 2015 *)

n=0; for (m = 1, 10^9, s=abs(moebius(sigma(m)*eulerphi(m))); if (s==1, write("b065299.txt", n++, " ", m); if (n==500, return))) \\ Harry J. Smith, Oct 15 2009

is(f)=my(n=#f~, v=List()); for(i=1,n, if(f[i,1]>2, listput(v,f[i,1]-1)); if(f[i,2]>2, return(0), f[i,2]>1, listput(v,f[i,1])); listput(v, (f[i,1]^(f[i,2]+1)-1)/(f[i,1]-1))); for(i=2,#v, for(j=1,i-1, if(gcd(v[i],v[j])>1, return(0)))); for(i=1,#v, if(!issquarefree(v[i]), return(0))); 1
sq(f)=f[,2]*=2; f
double(f)=if(#f~ && f[1,1]==2, f[1,2]++, f=concat([2,1],f)); f
list(lim)=my(v=List()); forsquarefree(n=1,sqrtint(lim\1), if(is(sq(n[2])), listput(v,n[1]^2))); forsquarefree(n=1,sqrtint(lim\2), if(is(double(sq(n[2]))), listput(v,2*n[1]^2))); Set(v) \\ Charles R Greathouse IV, Feb 05 2018

Solutions to abs(A008683(A000203(x)*A000010(x))) = 1.

A069249 a(n) = n^2 - phi(n)*sigma(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 12, 1, 4, 3, 28, 1, 32, 1, 52, 33, 8, 1, 90, 1, 64, 57, 124, 1, 96, 5, 172, 9, 112, 1, 324, 1, 16, 129, 292, 73, 204, 1, 364, 177, 160, 1, 612, 1, 256, 153, 532, 1, 320, 7, 640, 297, 352, 1, 756, 145, 256, 369, 844, 1, 912, 1, 964, 225, 32, 193, 1476, 1, 592

Offset: 1

Benoit Cloitre, Apr 13 2002

Always >0 for n>0. a(n)=1 if n is prime.

If p is a prime and k is a natural number then a(p^k)=p^(k-1) because a(p^k)=(p^k)^2-sigma(p^k)*phi(p^k) =p^(2k)-(p-1)*p^(k-1)*(p^(k+1)-1)/(p-1)=p^(k-1). If n is a composite number then a(n)>1 and a(1)=0, so n is prime iff a(n)=1. - Farideh Firoozbakht, Nov 15 2005

			sigma(10) = 18; phi(10) = 4; 10^2 - sigma(10)*phi(10) = 28. sigma(p) = p+1; phi(p) = p-1; p^2 - (p+1)(p-1) = 1. [From _Walter Nissen_, Aug 29 2009]

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A065465, A164875, A164876.

Table[n^2-EulerPhi[n]DivisorSigma[1,n],{n,70}] (* Harvey P. Dale, Oct 22 2016 *)

a(n)=n^2-eulerphi(n)*sigma(n) \\ Charles R Greathouse IV, Nov 27 2013

a(n) = n^2-A062354(n). - R. J. Mathar, Oct 01 2011

Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 1 - A065465 = 0.118486... . - Amiram Eldar, Dec 04 2023

cp's OEIS Frontend

A306724 Least number k > 1 such that A062354(k) is an n-th power, where A062354 is the product of sigma (A000203) and phi (A000010).

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A014574 Average of twin prime pairs.

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A065465 Decimal expansion of Product_{p prime} (1 - 1/(p^2*(p+1))).

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A062355 a(n) = d(n) * phi(n), where d(n) is the number of divisors function.

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A064840 a(n) = tau(n)*sigma(n).

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A330523 Decimal expansion of Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4).

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