A020488 -id:A020488 - cp's OEIS frontend

A079541 Numbers n such that phi(n) = d(n).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jan 23 2003

dead

Duplicate of A020488. - Peter Luschny, Aug 26 2011

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 11.

Cf. A000010, A000005, A063070.

Select[Range[500],EulerPhi[#]==DivisorSigma[0,#]&] (* Harvey P. Dale, Jun 18 2011 *)

Corrected by Harvey P. Dale, Jun 18 2011

A308664 Numbers k such that tau(k) and phi(k) are the legs of a Pythagorean triple.

Original entry on oeis.org

20, 36, 60, 100, 300

Offset: 1

Antonio Roldán, Jul 14 2019

The sequence is finite since for all large enough n, we have tau(n) < n^(1/4) and phi(n) > n^(3/4) while, if x < y are the legs of a Pythagorean triangle, we always have y < x^2/2. - Giovanni Resta, Jul 27 2019

From Resta's inequality it can be deduced that phi(n) <= 2304. Then it's easy to see that the sequence is full. - Max Alekseyev, May 30 2024

			60 is in this sequence because tau(60) = 12 and phi(60) = 16, legs of the Pythagorean triple {12, 16, 20} (12^2 + 16^2 = 20^2).

Cf. A000005, A000010, A020488, A062516, A063469, A063470, A112954.

Select[Range[300], IntegerQ@Sqrt[DivisorSigma[0, #]^2 + EulerPhi[#]^2] &] (* Amiram Eldar, Jul 26 2019 *)

for(i = 1,  2000, a = eulerphi(i); b = numdiv(i); if(issquare(a^2 + b^2), print1(i,", ")))

"full" keyword added by Max Alekseyev, May 30 2024

A378089 Irregular triangle read by rows in which row n lists the numbers k such that phi(k)/tau(k) = n.

Original entry on oeis.org

1, 3, 8, 10, 18, 24, 30, 5, 9, 15, 28, 40, 72, 84, 90, 120, 7, 21, 26, 56, 70, 78, 108, 126, 168, 210, 34, 45, 52, 102, 140, 156, 252, 360, 420, 11, 33, 88, 110, 198, 264, 330, 13, 35, 39, 63, 76, 104, 105, 130, 228, 234, 280, 312, 390, 504, 540, 630, 840, 58, 98, 174, 294

Offset: 1

Mohammed Yaseen, Nov 16 2024

			Triangle begins:
  n=1: 1, 3, 8, 10, 18, 24, 30;
  n=2: 5, 9, 15, 28, 40, 72, 84, 90, 120;
  n=3: 7, 21, 26, 56, 70, 78, 108, 126, 168, 210;
  n=4: 34, 45, 52, 102, 140, 156, 252, 360, 420;
  n=5: 11, 33, 88, 110, 198, 264, 330;
  n=6: 13, 35, 39, 63, 76, 104, 105, 130, 228, 234, 280, 312, 390, 504, 540, 630, 840;
  n=7: 58, 98, 174, 294;
  ...

Cf. A000005 (tau), A000010 (phi).
Cf. A020488 (row 1), A062516 (row 2), A063469 (row 3), A063470 (row 4).
Cf. A112954 (row lengths), A175667 (1st column), A112955 (right column), A020491 (ordered terms).

cp's OEIS Frontend

A079541 Numbers n such that phi(n) = d(n).

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A308664 Numbers k such that tau(k) and phi(k) are the legs of a Pythagorean triple.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A378089 Irregular triangle read by rows in which row n lists the numbers k such that phi(k)/tau(k) = n.

Views

Author

Keywords

Examples

Crossrefs