A110086 -id:A110086 - cp's OEIS frontend

A051612 a(n) = sigma(n) - phi(n).

0, 2, 2, 5, 2, 10, 2, 11, 7, 14, 2, 24, 2, 18, 16, 23, 2, 33, 2, 34, 20, 26, 2, 52, 11, 30, 22, 44, 2, 64, 2, 47, 28, 38, 24, 79, 2, 42, 32, 74, 2, 84, 2, 64, 54, 50, 2, 108, 15, 73, 40, 74, 2, 102, 32, 96, 44, 62, 2, 152, 2, 66, 68, 95, 36, 124, 2, 94, 52, 120, 2, 171, 2, 78

Offset: 1

Jud McCranie

nonn

			a(4) = sigma(4) - phi(4) = 7-2 = 5.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000 (First 1000 terms from T. D. Noe.)

Cf. A000010, A000203, A110085, A110086, A110087.

Cf. A240960, A292769 (partial sums).

a051612 n = a000203 n - a000010 n -- Reinhard Zumkeller, Aug 05 2014

a[n_] := SeriesCoefficient[Sum[(1-MoebiusMu[k])*x^k/(1-x^k)^2, {k, 1, n}], {x, 0, n}]; Array[a, 74] (* Jean-François Alcover, Sep 29 2017, after Ilya Gutkovskiy *)

a(n)=sigma(n)-eulerphi(n) \\ Charles R Greathouse IV, May 09 2013

a(n) = A000203(n)-A000010(n).
a(n) > 1 for n > 1; a(n) = 2 if and only if n is prime. - Charles R Greathouse IV, May 09 2013
G.f.: Sum_{k>=1} (1 - mu(k))*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Sep 29 2017

A110088 a(n) = tau(n)^omega(n), where tau = A000005 and omega = A001221.

Original entry on oeis.org

1, 2, 2, 3, 2, 16, 2, 4, 3, 16, 2, 36, 2, 16, 16, 5, 2, 36, 2, 36, 16, 16, 2, 64, 3, 16, 4, 36, 2, 512, 2, 6, 16, 16, 16, 81, 2, 16, 16, 64, 2, 512, 2, 36, 36, 16, 2, 100, 3, 36, 16, 36, 2, 64, 16, 64, 16, 16, 2, 1728, 2, 16, 36, 7, 16, 512, 2, 36, 16, 512, 2, 144, 2, 16, 36, 36, 16

Offset: 1

Reinhard Zumkeller, Jul 11 2005

a(n) depends only on the prime signature of n.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A001221, A110085, A110086, A110087, A240960.

a110088 n = a000005 n ^ a001221 n  -- Reinhard Zumkeller, Aug 05 2014

a[n_] := DivisorSigma[0, n]^PrimeNu[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2024 *)

a(n) = {my(f = factor(n)); numdiv(f)^omega(f);} \\ Amiram Eldar, Sep 15 2024

a(n) = A000005(n)^A001221(n).

A110085 Numbers k such that sigma(k) - phi(k) < tau(k)^omega(k), where sigma = A000203, phi = A000010, tau = A000005 and omega = A001221.

Original entry on oeis.org

1, 6, 10, 12, 18, 20, 24, 30, 36, 42, 60, 66, 70, 78, 84, 90, 102, 105, 110, 114, 120, 126, 130, 132, 138, 140, 150, 154, 156, 165, 168, 170, 174, 180, 182, 186, 190, 195, 198, 204, 210, 220, 222, 228, 230, 231, 234, 238, 240, 246, 252, 255, 258, 260, 264, 266

Offset: 1

Reinhard Zumkeller, Jul 11 2005

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Intersection of A018252 and A110086

Cf. A000005, A000010, A000203, A001221, A051612, A110087, A110088.

a110085 n = a110085_list !! (n-1)
a110085_list = filter (\x -> a051612 x < a110088 x) [1..]
-- Reinhard Zumkeller, Aug 05 2014

q[k_] := DivisorSigma[1, k] - EulerPhi[k] < DivisorSigma[0, k]^PrimeNu[k]; Select[Range[300], q] (* Amiram Eldar, Sep 15 2024 *)

is(n)=sigma(n)-eulerphi(n)Charles R Greathouse IV, Feb 14 2013

A051612(a(n)) < A110088(a(n)).

A110087 Numbers k such that sigma(k) - phi(k) > tau(k)^omega(k), where sigma = A000203, phi = A000010, tau = A000005 and omega = A001221.

Original entry on oeis.org

4, 8, 9, 14, 16, 21, 22, 25, 26, 27, 28, 32, 33, 34, 35, 38, 39, 40, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 64, 65, 68, 69, 72, 74, 75, 76, 77, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 104, 106, 108, 111, 112, 115, 116, 117, 118

Offset: 1

Reinhard Zumkeller, Jul 11 2005

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Complement of A110086.

Cf. A000005, A000010, A000203, A001221, A051612, A110088.

a110087 n = a110087_list !! (n-1)
a110087_list = filter (\x -> a051612 x > a110088 x) [1..]
-- Reinhard Zumkeller, Aug 05 2014

q[k_] := DivisorSigma[1, k] - EulerPhi[k] > DivisorSigma[0, k]^PrimeNu[k]; Select[Range[120], q] (* Amiram Eldar, Sep 15 2024 *)

is(n)=sigma(n)-eulerphi(n)>numdiv(n)^omega(n) \\ Charles R Greathouse IV, Feb 19 2013

A051612(a(n)) > A110088(a(n)).

A240960 Numbers m such that sigma(m) - phi(m) = tau(m)^omega(m), where sigma=A000203, phi=A000010, tau=A000005 and omega=A001221.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset: 1

Reinhard Zumkeller, Aug 05 2014

nonn

a(n) = A182140(n) for n <= 35.

All primes p are in the sequence since (p+1) - (p-1) = 2^1. The first composites are 15, 119748396, 139254850, 187768485, 1420027536, 3991789984. A182140 seems unrelated. - Jens Kruse Andersen, Aug 05 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A000203, A001221, A051612, A110088, A182140, A110086.

a240960 n = a240960_list !! (n-1)
a240960_list = filter (\x -> a051612 x == a110088 x) [1..]

with(numtheory):
filter:= n -> sigma(n) - phi(n) = tau(n)^nops(factorset(n)):
select(filter, [$1..1000]); # Robert Israel, Aug 05 2014

Select[Range[300], DivisorSigma[1, #] - EulerPhi[#] == DivisorSigma[0, #]^PrimeNu[#]&] (* Jean-François Alcover, Mar 08 2019 *)

is(n)=my(f=factor(n)); sigma(f)-eulerphi(f)==numdiv(f)^omega(f) \\ Charles R Greathouse IV, Nov 26 2014

from sympy import totient,divisors,divisor_count,primefactors
filter(lambda x:sum(divisors(x))-totient(x)==divisor_count(x)**len(primefactors(x)), range(1,10**5)) # Chai Wah Wu, Aug 05 2014

A353276 a(n) = phi(n) + tau(n)^omega(n) - sigma(n).

Original entry on oeis.org

1, 0, 0, -2, 0, 6, 0, -7, -4, 2, 0, 12, 0, -2, 0, -18, 0, 3, 0, 2, -4, -10, 0, 12, -8, -14, -18, -8, 0, 448, 0, -41, -12, -22, -8, 2, 0, -26, -16, -10, 0, 428, 0, -28, -18, -34, 0, -8, -12, -37, -24, -38, 0, -38, -16, -32, -28, -46, 0, 1576, 0, -50, -32, -88, -20, 388, 0, -58, -36, 392, 0, -27, 0, -62, -48, -68, -20, 368

Offset: 1

Antti Karttunen, Apr 27 2022

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A000203, A001221, A051612, A110088.

Cf. A110087 (positions of negative terms), A110086 (of terms >= 0), A110085 (of terms > 0).

Array[#1 + #3^#2 - #4 & @@ Flatten@ {EulerPhi[#], PrimeNu[#], DivisorSigma[{0, 1}, #]} &, 78] (* Michael De Vlieger, Apr 27 2022 *)

A353276(n) = (eulerphi(n) + (numdiv(n)^omega(n)) - sigma(n));

a(n) = A110088(n) - A051612(n) = A000010(n) + A000005(n)^A001221(n) - A000203(n).

a(p) = 0 for all primes p.

cp's OEIS Frontend

A051612 a(n) = sigma(n) - phi(n).

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A110088 a(n) = tau(n)^omega(n), where tau = A000005 and omega = A001221.

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A110085 Numbers k such that sigma(k) - phi(k) < tau(k)^omega(k), where sigma = A000203, phi = A000010, tau = A000005 and omega = A001221.

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A110087 Numbers k such that sigma(k) - phi(k) > tau(k)^omega(k), where sigma = A000203, phi = A000010, tau = A000005 and omega = A001221.

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A240960 Numbers m such that sigma(m) - phi(m) = tau(m)^omega(m), where sigma=A000203, phi=A000010, tau=A000005 and omega=A001221.

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A353276 a(n) = phi(n) + tau(n)^omega(n) - sigma(n).

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