The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A341939 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A341939

Numbers m such that phi(m)/tau(m) is a square of an integer where phi is the Euler totient function (A000010) and tau is the number of divisors function (A000005).
(history; published version)

#20 by N. J. A. Sloane at Sat Feb 27 21:57:13 EST 2021

STATUS

proposed

approved

#19 by Bernard Schott at Sat Feb 27 11:21:33 EST 2021

STATUS

editing

proposed

#18 by Bernard Schott at Sat Feb 27 11:21:13 EST 2021

COMMENTS

-> The seven terms that satisfy also tau(m) = phi(m) form the subsequence A020488 with phi(m)/tau(m) = 1^2.

-> Primes p of the form 2*k^2 + 1 (A090698) form another subsequence because tau(p) = 2 and phi(p) = p-1 = 2*k^2, so phi(p)/tau(p) = k^2.

-> Cubes p^3 where p is a prime of the form k^2+1 (A002496) form another subset because if p = 2, phi(8)/tau(8)=1, and if p odd, phi(p^3)/tau(p^3) = (k*p/2)^2 with k even.

STATUS

proposed

editing

#17 by Bernard Schott at Wed Feb 24 13:05:13 EST 2021

STATUS

editing

proposed

#16 by Bernard Schott at Wed Feb 24 13:03:15 EST 2021

MAPLE

with(numtheory): filter:= q -> phi(q)/tau(q) = floor(phi(q)/tau(q)) and issqr(phi(q)/tau(q)) : select(filter, [$1..750]);

CROSSREFS

Subsequences: A090698, A020488.

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

STATUS

proposed

editing

Discussion

Wed Feb 24

13:05

Bernard Schott: Added with(numtheory) + xrefs.

#15 by Bernard Schott at Wed Feb 24 05:45:39 EST 2021

STATUS

editing

proposed

Discussion

Wed Feb 24

11:26

Michel Marcus: 18/8 is 9/4 = (3/2)^2 a square , but not an integer

11:40

Bernard Schott: Yes Michel, I know; in French "un carré parfait" is "le carré d'un entier"; it seems to be the same in English, auquel cas: perfect square = square of an integer;  but it is clearer write square of an integer, thanks.

#14 by Bernard Schott at Wed Feb 24 05:44:55 EST 2021

NAME

Numbers m such that phi(m)/tau(m) is a perfect square of an integer where phi is the Euler totient function (A000010) and tau is the number of divisors function (A000005).

Discussion

Wed Feb 24

05:45

Bernard Schott: Ok, added integer in the Name.

#13 by Michel Marcus at Wed Feb 24 04:19:25 EST 2021

PROG

(PARI) isok(m) = issquaremy(x=eulerphi(m)/numdiv(m)); (denominator(x)==1) && issquare(x); \\ Michel Marcus, Feb 24 2021

#12 by Michel Marcus at Wed Feb 24 04:02:32 EST 2021

PROG

(PARI) isok(m) = issquare(eulerphi(m)/numdiv(m)); \\ Michel Marcus, Feb 24 2021

STATUS

proposed

editing

Discussion

Wed Feb 24

04:06

Michel Marcus: eulerphi(54)/numdiv(54) = 9/4 a square  = (3/2) ^2

04:08

Michel Marcus: name did not say integer ??

04:11

Bernard Schott: I do not think: phi(54)  = 18 and tau(54) = 8, and 18/8 is not perfect square. 54 is the first counterexample proposed in first comment, it will be also first term of A341940.

04:14

Bernard Schott: A perfect square = an integer that is the square of an integer in Wiki here: https://en.wikipedia.org/wiki/Square_number . Please, do you think it is necessary add integer?

#11 by Amiram Eldar at Wed Feb 24 03:51:06 EST 2021

STATUS

editing

proposed

Discussion

Wed Feb 24

04:01

Michel Marcus: 54 has been missed ?