A112587 -id:A112587 - cp's OEIS frontend

A020488 Numbers n such that tau(n) (or sigma_0(n)) = phi(n).

1, 3, 8, 10, 18, 24, 30

Offset: 1

Numbers satisfying A000005(n) = A000010(n).
This sequence is complete because tau(n) < n^(2/3) for all n except a few small numbers, whereas phi(n) > n/(exp(gamma) * log(log(n)) + 3/(log(log(n))) for n > 2. log(log(n)) grows slowly, so phi(n) > tau(n) for all n greater than some relatively small constant. - Jud McCranie, Jun 17 2005
Subset of A112587. - Reinhard Zumkeller, Sep 14 2005
A. P. Minin proved in 1894 that these are the only terms. - Amiram Eldar, May 14 2017

			10 has four divisors: 1, 2, 5, 10, so tau(10) = 4. And four numbers less than 10 are coprime to 10: 1, 3, 7, 9, so phi(10) = 4. Since tau(10) = phi(10), 10 is in the sequence.
phi(12) = 4 also, but 12 has more than four divisors: 1, 2, 3, 4, 6, 12. So 12 is not in the sequence.

L. E. Dickson, History of the Theory of Numbers, Vol. 1, (1919), Chapter X, p. 313.
Jean-Marie De Koninck, Those Fascinating Numbers, translated by the author. Providence, Rhode Island (2009) American Mathematical Society, p. 3.
G. Pólya and G. Szegő, Problems and Theorems in Analysis II, Springer, 1976, Part VIII, Problem 45.

A. P. Minin, On integers N such that the number of divisors of N equals the number of integers less than N and prime to it, Math. Soc. Moscow, Vol. 17, (1894), pp. 537-544 (some front matter is in English and German, article is in Russian)

Cf. A064374, A064375, A064376, A064377, A000005, A000010.

Cf. A112954, A062516, A063469, A063470.

Filtered([1..1000],n->Tau(n)=Phi(n)); # Muniru A Asiru, Dec 20 2018

[n: n in [1..1000] | EulerPhi(n) eq NumberOfDivisors(n)]; // Marius A. Burtea, Dec 20 2018

select(k->tau(k)=phi(k),[$1..1000]); # Peter Luschny, Aug 26 2011

k = 1; s = Select[Range[100000], Equal[Sign[DivisorSigma[k - 1, #] - EulerPhi[#]^k ], 0 ] &]
Select[Range[1000], DivisorSigma[0, #] == EulerPhi[#] &] (* Alonso del Arte, Jan 15 2019 *)

isok(n) = numdiv(n) == eulerphi(n); \\ Michel Marcus, May 14 2017

A112588 Numbers m such that phi(m) > 2*tau(m), where phi = A000010 and tau = A000005.

Original entry on oeis.org

7, 11, 13, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96

Offset: 1

Reinhard Zumkeller, Sep 14 2005

nonn

Complement of A112587.

Eric Weisstein's World of Mathematics, Totient Function
Eric Weisstein's World of Mathematics, Divisor Function

Cf. A000005, A000010, A112587.

Select[Range[96], EulerPhi[#]>2DivisorSigma[0,#]&] (* James C. McMahon, Jun 04 2024 *)

a(n) = n + 27 for n >= 94. - Jianing Song, Jul 28 2018

A248888 Numbers k such that prime(k) <= 2*sigma(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 28, 30, 36, 40, 42, 48, 60, 72, 84, 90, 96, 120, 180

Offset: 1

Farideh Firoozbakht, Oct 17 2014

Obviously only for n = 1, prime(n) = 2*sigma(n).

It is interesting that all terms of the sequence A112587 are in the sequence and this sequence has only two more terms 96 & 180. Hence if phi(n) <= 2*tau(n) then prime(n) <= 2*sigma(n).

Cf, A000005, A000010, A000040, A000203.

Select[Range[200], Prime[#] <= 2DivisorSigma[1,#]&]

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A020488 Numbers n such that tau(n) (or sigma_0(n)) = phi(n).

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A112588 Numbers m such that phi(m) > 2*tau(m), where phi = A000010 and tau = A000005.

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A248888 Numbers k such that prime(k) <= 2*sigma(k).

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