A078148 -id:A078148 - cp's OEIS frontend

A033632 Numbers k such that sigma(phi(k)) = phi(sigma(k)).

1, 9, 225, 242, 516, 729, 3872, 13932, 14406, 17672, 18225, 20124, 21780, 29262, 29616, 45996, 65025, 76832, 92778, 95916, 106092, 106308, 114630, 114930, 121872, 125652, 140130, 140625, 145794, 149124, 160986, 179562, 185100, 234876, 248652, 252978, 256860

Offset: 1

Jud McCranie

The largest term of this sequence that I found is 3^9550. Also, if (1/2)*(3^(k+1)-1) is prime (k+1 is a term of A028491) then 3^k is in the sequence, namely sigma(phi(3^k)) = phi(sigma(3^k)) (the proof is easy). - Farideh Firoozbakht, Feb 09 2005

R. K. Guy, Unsolved Problems in Number Theory, 2nd edition, Springer Verlag, 1994, section B42, p. 99.

Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 200 terms from T. D. Noe)
S. W. Golomb, Equality among number-theoretic functions, Unpublished manuscript. (Annotated scanned copy)
Walter Nissen, sigma(phi(n)) = phi(sigma(n)), Up for the Count !
Walter Nissen, sigma(phi(n)) = phi(sigma(n)): From "5" to "5 figures", Up for the Count !, Nov. 14, 2000
Walter Nissen, Addendum to : sigma(phi()): From "5" to "5 figures", Up for the Count !, June 8, 2008
Walter Nissen, Elaboration of : sigma(phi()): From "5" to "5 figures", Up for the Count !, Oct. 15, 2010

Cf. A000203, A000010, A028491, A078148.

a033632 n = a033632_list !! (n-1)
a033632_list = filter (\x -> a062401 x == a062402 x) [1..]
-- Reinhard Zumkeller, Jan 04 2013

Select[ Range[ 10^6 ], DivisorSigma[ 1, EulerPhi[ # ] ] == EulerPhi[ DivisorSigma[ 1, # ] ] & ]

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

from sympy import divisor_sigma as sigma, totient as phi
def ok(n): return sigma(phi(n)) == phi(sigma(n))
def aupto(nn): return [m for m in range(1, nn+1) if ok(m)]
print(aupto(10**4)) # Michael S. Branicky, Jan 09 2021

A062401(a(n)) = A062402(a(n)). - Reinhard Zumkeller, Jan 04 2013

A385122 a(n) = d(phi(n)) - phi(d(n)) where d(n) = A000005(n) is the number of divisors and phi(n) = A000010(n) is the Euler totient function.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 1, 2, 1, 3, 1, 5, 2, 2, 0, 4, 2, 5, 2, 4, 2, 3, 0, 4, 4, 4, 4, 5, 0, 7, 3, 4, 3, 6, 0, 8, 4, 6, 1, 7, 2, 7, 4, 6, 2, 3, 1, 6, 4, 4, 6, 5, 2, 6, 4, 7, 4, 3, 1, 11, 6, 7, 0, 8, 2, 7, 4, 4, 4, 7, 4, 11, 7, 6, 7, 10, 4, 7, 2, 4, 6, 3, 4, 5, 6

Offset: 1

Sean A. Irvine, Jun 18 2025

sign

First negative value is a(120) = -2.

Sean A. Irvine, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A062821, A078148, A078150, A163109.

A385122[n_] := DivisorSigma[0, EulerPhi[n]] - EulerPhi[DivisorSigma[0, n]];
Array[A385122, 100] (* Paolo Xausa, Jun 19 2025 *)

a(n) = numdiv(eulerphi(n)) - eulerphi(numdiv(n)); \\ Michel Marcus, Jun 19 2025

a(n) = A000005(A000010(n)) - A000010(A000005(n)).

a(n) = A062821(n) - A163109(n).

A378315 Odd numbers k such that d(phi(k)) = phi(d(k)), where d=A000005 and phi=A000010.

Original entry on oeis.org

1, 15230439315, 18887708385, 74989937295, 78103226565, 86031400455, 114958521405, 179883837315, 210096608085, 367588711035, 418094581905, 461441147895, 590648954805, 649146021615, 685787041485, 836850895335, 874197762165, 990695282031, 996070731201, 1002913997085, 1016370465201, 1029306324501, 1029869788311, 1039854060045, 1043905592457

Offset: 1

Max Alekseyev, Jan 09 2025

nonn

For n > 1, we have A001222(a(n)) >= 9. The smallest a(n) with A001222(a(n)) = 9 is a(65) = 1244586078645.

Max Alekseyev, Table of n, a(n) for n = 1..65
Tong Lingling et al., Find the smallest odd integer n>1 such that phi(tau(n)) = tau(phi(n)), MathOverflow, 2025.
zhouzixiang2004, DMOPC'21 Contest 10 P5 - Number Theory, DMOJ Monthly Open Programming Competition, 2021.

Subsequence of A078148.

Cf. A000005, A000010, A033632.

A218006 Numbers n such that sigma(tau(phi(n))) = tau(phi(sigma(n))) = phi(sigma(tau(n))).

Original entry on oeis.org

1, 34, 36, 96, 128, 468, 1200, 21216, 102060, 110976, 117684, 211428, 331380, 366660, 437220, 511680, 530712, 706560, 710388, 726240, 732240, 759360, 838080, 845376, 875840, 911040, 975936, 1014016, 1041216, 1093440, 1110720, 1141440, 1167696, 1289280

Offset: 1

Jayanta Basu, Mar 26 2013

nonn

Here phi denotes Euler's totient function, tau(n) denotes number of divisors of n and sigma(n) denotes sum of all divisors of n. Only cyclic rotation of operators is considered.

Cf. A000005, A000010, A000203, A033632, A076361, A078148.

Select[Range[1000000], DivisorSigma[1, DivisorSigma[0, EulerPhi[#]]] == DivisorSigma[0, EulerPhi[DivisorSigma[1, #]]] == EulerPhi[DivisorSigma[1, DivisorSigma[0, #]]] &]

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A033632 Numbers k such that sigma(phi(k)) = phi(sigma(k)).

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A385122 a(n) = d(phi(n)) - phi(d(n)) where d(n) = A000005(n) is the number of divisors and phi(n) = A000010(n) is the Euler totient function.

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A378315 Odd numbers k such that d(phi(k)) = phi(d(k)), where d=A000005 and phi=A000010.

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A218006 Numbers n such that sigma(tau(phi(n))) = tau(phi(sigma(n))) = phi(sigma(tau(n))).

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