A062355 -id:A062355 - cp's OEIS frontend

A336745 Numbers m that divide the product phi(m) * sigma(m) * tau(m), where phi is the Euler totient function (A000010), sigma is the sum of divisors function (A000203) and tau is the number of divisors function (A000005).

1, 2, 6, 8, 9, 12, 18, 24, 28, 32, 36, 40, 54, 72, 80, 84, 96, 108, 117, 120, 128, 135, 144, 162, 196, 200, 216, 224, 234, 240, 243, 252, 270, 288, 324, 360, 384, 400, 405, 448, 468, 486, 496, 512, 540, 576, 588, 600, 625, 640, 648, 672, 675, 720, 756, 768, 775, 810, 819

Offset: 1

Bernard Schott, Aug 02 2020

nonn

If s and t are terms with gcd(s, t) = 1, then s*t is another term as phi, sigma and tau are multiplicative functions.

The only prime term is 2 because prime p must divide 2*(p-1)*(p+1) to be a term.

			For 24, phi(24) = 8, sigma(24) = 60 and tau(24) = 8, then 8*60*8 / 24 = 160, hence 24 is a term.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A000203, A062355.

Subsequences: A000396 (perfect numbers), A005820 (tri-perfect), A027687 (4-perfect), A046060 (5-multiperfect), A046061 (6-multiperfect), A007691 (multiply-perfect numbers), A336715 (m divides phi(m)*tau(m)), A004171, A005010.

with(numtheory):
filter:= m -> irem(tau(m)*phi(m)*sigma(m), m) =0:
select(filter,[$1..850]);

Select[Range[1000], Divisible[Times @@ DivisorSigma[{0, 1}, #] * EulerPhi[#], #] &] (* Amiram Eldar, Aug 02 2020 *)

isok(m) = !(eulerphi(m)*sigma(m)*numdiv(m) % m); \\ Michel Marcus, Aug 05 2020

A062816 a(n) = phi(n)tau(n) - 2n = A000010(n)A000005(n) - 2*n.

Original entry on oeis.org

-1, -2, -2, -2, -2, -4, -2, 0, 0, -4, -2, 0, -2, -4, 2, 8, -2, 0, -2, 8, 6, -4, -2, 16, 10, -4, 18, 16, -2, 4, -2, 32, 14, -4, 26, 36, -2, -4, 18, 48, -2, 12, -2, 32, 54, -4, -2, 64, 28, 20, 26, 40, -2, 36, 50, 80, 30, -4, -2, 72, -2, -4, 90, 96, 62, 28, -2, 56, 38, 52, -2, 144, -2, -4, 90, 64, 86, 36, -2, 160, 108, -4, -2, 120, 86, -4

Offset: 1

Labos Elemer, Jul 20 2001

sign

It can be shown that phi(n)*tau(n) >= n, which means that quotient = n/tau(n) <= phi(n); note: a(n)+5 is positive.

The value is always positive except when a(n) = 0 for {8,9,12}; or a(n) = -2 for primes together with 4 (i.e., for A046022 but without 1); or a(n) = -4 for A001747 (without 2 and 4); or a(n) = -1 for n = 1.

Harry J. Smith, Table of n, a(n) for n = 1..2000

Cf. A000010, A000005, A062355, A062516, A046022, A001747, A036762, A036763, A051728, A051729, A051730.

Table[EulerPhi[n]DivisorSigma[0,n]-2n,{n,90}] (* Harvey P. Dale, Feb 03 2021 *)

a(n)={eulerphi(n)*numdiv(n) - 2*n} \\ Harry J. Smith, Aug 11 2009

a(n) = A062355(n) - 2*n. - Amiram Eldar, Jul 10 2024

Offset changed from 0 to 1 by Harry J. Smith, Aug 11 2009

A094181 a(n) = (n - tau(n))*(n - phi(n)), where tau=A000005 and phi=A000010.

Original entry on oeis.org

0, 0, 1, 2, 3, 8, 5, 16, 18, 36, 9, 48, 11, 80, 77, 88, 15, 144, 17, 168, 153, 216, 21, 256, 110, 308, 207, 352, 27, 484, 29, 416, 377, 540, 341, 648, 35, 680, 525, 768, 39, 1020, 41, 912, 819, 1008, 45, 1216, 322, 1320, 893, 1288, 51, 1656, 765, 1536, 1113, 1620

Offset: 1

Reinhard Zumkeller, May 06 2004

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A062355.

[(n-NumberOfDivisors(n))*(n-EulerPhi(n)): n in [1..60]]; // Vincenzo Librandi, Aug 31 2018

Array[(#-DivisorSigma[0,#])(#-EulerPhi[#])&,60] (* Harvey P. Dale, May 18 2012 *)

a(n) = A049820(n)*A051953(n).

A110601 a(n) = phi(n)*tau(n)^2, where phi is Euler's totient function and tau(n) is the number of divisors of n.

Original entry on oeis.org

1, 4, 8, 18, 16, 32, 24, 64, 54, 64, 40, 144, 48, 96, 128, 200, 64, 216, 72, 288, 192, 160, 88, 512, 180, 192, 288, 432, 112, 512, 120, 576, 320, 256, 384, 972, 144, 288, 384, 1024, 160, 768, 168, 720, 864, 352, 184, 1600, 378, 720, 512, 864, 208, 1152, 640

Offset: 1

Emeric Deutsch, Jul 29 2005

			a(4)=18 because phi(4)=2 and tau(4)=3.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Stefan Porubsky and M. G. Greening, Problem E2351, Amer. Math. Monthly, Vol. 80, No. 4 (1973), p. 436.

Cf. A000005, A000010, A035116, A062355.

[EulerPhi(n)*NumberOfDivisors(n)^2: n in [1..60]]; // Vincenzo Librandi, Jun 21 2017

with(numtheory): a:=n->phi(n)*tau(n)^2: seq(a(n),n=1..70);

Table[EulerPhi[n]DivisorSigma[0,n]^2,{n,60}] (* Harvey P. Dale, Nov 29 2011 *)

a(n) = eulerphi(n)*numdiv(n)^2; \\ Michel Marcus, Jun 21 2017

a(n) = A000010(n) * A035116(n) = A062355(n) * A000005(n). - R. J. Mathar, Jul 26 2022

Multiplicative with a(p^e) = (e+1)^2*(p-1)*p^(e-1). - Amiram Eldar, Dec 29 2022

A127528 Triangle T(n,k) read by rows: tau(n)*phi(n/k) if k|n, else 0.

Original entry on oeis.org

1, 2, 2, 4, 0, 2, 6, 3, 0, 3, 8, 0, 0, 0, 2, 8, 8, 4, 0, 0, 4, 12, 0, 0, 0, 0, 0, 2, 16, 8, 0, 4, 0, 0, 0, 4, 18, 0, 6, 0, 0, 0, 0, 0, 3, 16, 16, 0, 0, 4, 0, 0, 0, 0, 4, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 24, 12, 12, 12, 0, 6, 0, 0, 0, 0, 0, 6, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 24, 24

Offset: 1

Gary W. Adamson, Jan 17 2007

			First few rows of the triangle are:
1;
2, 2;
4, 0, 2;
6, 3, 0, 3;
8, 0, 0, 0, 2;
8, 8, 4, 0, 0, 4;
12, 0, 0, 0, 0, 0, 2;
...

Cf. A054523, A127527, A062355, A038040, A000005.

A127528 := proc(n,k) if n mod k = 0 then numtheory[tau](n)*numtheory[phi](n/k) ; else  0; end if; end proc:
seq(seq(A127528(n,k),k=1..n),n=1..15) ; # R. J. Mathar, Feb 05 2011

T(n,k) = A000005(n) * A054523(n,k).
T(n,1) = A062355(n).
T(n,n) = A000005(n).
sum_{k=1..n} T(n,k) = A038040(n).

A244342 a(n) = phi(n)*h(n) where phi() is the Euler totient function, A000010, and h() is A092089.

Original entry on oeis.org

1, 2, 6, 8, 12, 12, 18, 32, 30, 24, 30, 48, 36, 36, 72, 96, 48, 60, 54, 96, 108, 60, 66, 192, 100, 72, 126, 144, 84, 144, 90, 256, 180, 96, 216, 240, 108, 108, 216, 384, 120, 216, 126, 240, 360, 132, 138, 576, 210, 200, 288, 288, 156, 252, 360, 576, 324, 168

Offset: 1

Michel Marcus, Jun 26 2014

a(n) = Sum_{k=1..n} gcd(k^2-1, n) for those k that are coprime to n (see proof in link).

Multiplicative because both A000010 and A092089 are. - Andrew Howroyd, Jul 26 2018

Robert Israel, Table of n, a(n) for n = 1..10000
László Tóth, Menon's identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110 (see (36) in Corollary 15, p. 108); also on arXiv, arXiv:1103.5861 [math.NT], 2011.

Cf. A000010, A062355, A092089.

A244342:= proc(n) add(`if`(igcd(k,n)=1,igcd(k^2-1,n),0),k=1..n) end proc;
seq(A244342(i),i=1..1000); # Robert Israel, Jul 06 2014

h[n_] := Product[{p, e} = pe; Which[OddQ[p], 2 e + 1, p == 2 && e == 1, 2, True, 4 (e - 1)], {pe, FactorInteger[n]}]; h[1] = 1;
a[n_] := EulerPhi[n] h[n];
Array[a, 100] (* Jean-François Alcover, Apr 08 2020 *)

a(n) = sum(j=1, n, gcd(j^2-1,n)*(gcd(j,n)==1));

A318519 a(n) = A000005(n) * A003557(n).

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 16, 9, 4, 2, 12, 2, 4, 4, 40, 2, 18, 2, 12, 4, 4, 2, 32, 15, 4, 36, 12, 2, 8, 2, 96, 4, 4, 4, 54, 2, 4, 4, 32, 2, 8, 2, 12, 18, 4, 2, 80, 21, 30, 4, 12, 2, 72, 4, 32, 4, 4, 2, 24, 2, 4, 18, 224, 4, 8, 2, 12, 4, 8, 2, 144, 2, 4, 30, 12, 4, 8, 2, 80, 135, 4, 2, 24, 4, 4, 4, 32, 2, 36, 4, 12, 4, 4, 4, 192, 2, 42, 18, 90, 2, 8, 2, 32, 8

Offset: 1

Antti Karttunen, Sep 16 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A003557, A062355, A173557.

f[p_, e_] := (e + 1)*p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2023 *)

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A318519(n) = (numdiv(n)*A003557(n));

A318519(n) = { my(f=factor(n)); prod(i=1, #f~, (f[i,2]+1)*(f[i,1]^(f[i,2]-1))); };

Multiplicative with a(p^e) = (e+1)*(p^(e-1)).
a(n) = A000005(n) * A003557(n).
a(n) = A062355(n) / A173557(n).
Dirichlet g.f.: zeta(s-1)^2 * Product_{p prime} (1 - 2/p^(s-1) + 2/p^s - 1/p^(2*s-1) + 1/p^(2*s-2)). - Amiram Eldar, Sep 14 2023

A333557 a(n) = Sum_{d|n, gcd(d, n/d) = 1} uphi(d) * uphi(n/d), where uphi = unitary totient function (A047994).

Original entry on oeis.org

1, 2, 4, 6, 8, 8, 12, 14, 16, 16, 20, 24, 24, 24, 32, 30, 32, 32, 36, 48, 48, 40, 44, 56, 48, 48, 52, 72, 56, 64, 60, 62, 80, 64, 96, 96, 72, 72, 96, 112, 80, 96, 84, 120, 128, 88, 92, 120, 96, 96, 128, 144, 104, 104, 160, 168, 144, 112, 116, 192, 120, 120, 192, 126, 192, 160, 132, 192, 176, 192

Offset: 1

Ilya Gutkovskiy, Mar 26 2020

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

Cf. A001221, A029935, A034444, A047994, A055653, A062355, A145388.

uphi[1] = 1; uphi[n_] := Times @@ (#[[1]]^#[[2]] - 1 & /@ FactorInteger[n]); a[n_] := Sum[If[GCD[d, n/d] == 1, uphi[d] uphi[n/d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]
Table[Sum[If[GCD[d, n/d] == 1, (-2)^PrimeNu[n/d] 2^PrimeNu[d] d, 0], {d, Divisors[n]}], {n, 1, 70}]
f[p_, e_] := 2*(p^e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 30 2023 *)

a(n) = sumdiv(n, d, if (gcd(d, n/d) == 1, (-2)^omega(n/d)*2^omega(d)*d)); \\ Michel Marcus, Mar 27 2020

If n = Product (p_j^k_j) then a(n) = Product (2 * (p_j^k_j - 1)).
a(n) = 2^omega(n) * uphi(n).
a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-2)^omega(n/d) * 2^omega(d) * d.
a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * A145388(d).

A336715 Numbers m that divide the product phi(m) * tau(m), where tau is the number of divisors function (A000005) and phi is the Euler totient function (A000010).

Original entry on oeis.org

1, 2, 8, 9, 12, 18, 32, 36, 72, 80, 96, 108, 128, 144, 243, 288, 324, 400, 448, 486, 512, 576, 625, 720, 768, 864, 972, 1152, 1200, 1250, 1344, 1620, 1944, 2000, 2025, 2048, 2304, 2500, 2560, 2592, 2916, 3136, 3600, 3888, 4032, 4050, 4608, 5000, 5103, 5625, 6144, 6561, 6912

Offset: 1

Bernard Schott, Aug 01 2020

nonn

Numbers of the form q = 2^(2k+1) with k>=0 (A004171) form a subsequence because tau(q) * phi(q) / q = k + 1.

Numbers of the form q = 9 * 2^k with k>=0 (A005010) form another subsequence because tau(q) * phi(q) / q = k+1 (also).

			For 80, phi(80) = 32, tau(80) = 10 and tau(80)*phi(80)/80 = 4, hence 80 is a term.

David A. Corneth, Table of n, a(n) for n = 1..12173 (terms <= 10^15)

Cf. A000010 (phi), A000005 (tau), A062355.

Subsequences: A004171, A005010.

with(numtheory):
filter:= m-> irem(phi(m)*tau(m), m)=0:
select(filter, [$1..7000])[];

Select[Range[7000], Divisible[DivisorSigma[0, #] * EulerPhi[#], #] &] (* Amiram Eldar, Aug 01 2020 *)

isok(m) = (eulerphi(m)*numdiv(m) % m) == 0; \\ Michel Marcus, Aug 02 2020

A348061 a(n) = Sum_{k=1..n, gcd(n,k) = 1} n / gcd(n,k-1).

Original entry on oeis.org

1, 1, 4, 3, 16, 4, 36, 11, 34, 16, 100, 12, 144, 36, 64, 43, 256, 34, 324, 48, 144, 100, 484, 44, 396, 144, 304, 108, 784, 64, 900, 171, 400, 256, 576, 102, 1296, 324, 576, 176, 1600, 144, 1764, 300, 544, 484, 2116, 172, 1758, 396, 1024, 432, 2704, 304, 1600, 396, 1296, 784, 3364, 192

Offset: 1

Ilya Gutkovskiy, Sep 26 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A057660, A062355, A348060.

Table[Sum[If[GCD[n, k] == 1, n/GCD[n, k - 1], 0], {k, n}], {n, 60}]
f[p_, e_] := (p^(2 e + 1) - (p + 1) p^(2 e - 1) + 1)/(p + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 60]

a(n) = sum(k=1, n, if (gcd(n, k)==1, n/gcd(n, k-1))); \\ Michel Marcus, Sep 27 2021

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

Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} (1 - 1/p^2 - 1/(1 + p + p^2)) = 0.1381393084... . - Amiram Eldar, Nov 18 2022

cp's OEIS Frontend

A336745 Numbers m that divide the product phi(m) * sigma(m) * tau(m), where phi is the Euler totient function (A000010), sigma is the sum of divisors function (A000203) and tau is the number of divisors function (A000005).

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A062816 a(n) = phi(n)*tau(n) - 2n = A000010(n)*A000005(n) - 2*n.

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A094181 a(n) = (n - tau(n))*(n - phi(n)), where tau=A000005 and phi=A000010.

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A110601 a(n) = phi(n)*tau(n)^2, where phi is Euler's totient function and tau(n) is the number of divisors of n.

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A127528 Triangle T(n,k) read by rows: tau(n)*phi(n/k) if k|n, else 0.

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A244342 a(n) = phi(n)*h(n) where phi() is the Euler totient function, A000010, and h() is A092089.

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A318519 a(n) = A000005(n) * A003557(n).

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A333557 a(n) = Sum_{d|n, gcd(d, n/d) = 1} uphi(d) * uphi(n/d), where uphi = unitary totient function (A047994).

A062816 a(n) = phi(n)tau(n) - 2n = A000010(n)A000005(n) - 2*n.