The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A341938 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A341938

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

#22 by N. J. A. Sloane at Sat Feb 27 21:56:34 EST 2021

STATUS

proposed

approved

#21 by Bernard Schott at Sat Feb 27 03:05:16 EST 2021

STATUS

editing

proposed

#20 by Bernard Schott at Sat Feb 27 03:04:59 EST 2021

COMMENTS

-> The seven terms that satisfy tau(m) = phi(m) form the subsequence A020488.

-> 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 tau(p)*phi(p) = (2*k)^2.

-> Cubes p^3 where p is a prime of the form k^2+1 (A002496) form another subset with tau(p^3)*phi(p^3) = (2*k*p)^2.

#19 by Bernard Schott at Sat Feb 27 03:02:51 EST 2021

COMMENTS

The first 11 terms of this sequence are also the first 11 terms of A341939: m such that phi(m)/tau(m) is a perfect the square of an integer. Indeed, if phi(m)/tau(m) is a perfect square then phi(m)*tau(m) is also a square, but the converse is false. These counterexamples are in A341940, the first one is a(12) = 54.

STATUS

proposed

editing

Discussion

Sat Feb 27

03:04

Bernard Schott: In comment, replaced perfect number by square of an integer.

#18 by Jon E. Schoenfield at Wed Feb 24 22:41:27 EST 2021

STATUS

editing

proposed

#17 by Jon E. Schoenfield at Wed Feb 24 22:41:25 EST 2021

COMMENTS

The seven terms that satisfy tau(m) = phi(m) form the subsequence A020488.

STATUS

proposed

editing

#16 by Bernard Schott at Wed Feb 24 13:08:56 EST 2021

STATUS

editing

proposed

#15 by Bernard Schott at Wed Feb 24 13:08:21 EST 2021

MAPLE

with(numtheory): filter:= n -> issqr(phi(n)*tau(n)) : select(filter, [$1..750]);

CROSSREFS

Subsequences: A020488, A090698.

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

STATUS

proposed

editing

Discussion

Wed Feb 24

13:08

Bernard Schott: Added with(numtheory) + xrefs.

#14 by Michel Marcus at Wed Feb 24 04:05:00 EST 2021

STATUS

editing

proposed

#13 by Michel Marcus at Wed Feb 24 04:04:42 EST 2021

PROG

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

STATUS

proposed

editing