A070305 Numbers m such that Card(k>0 : phi(k)=phi(m)) = tau(m).

2, 4, 8, 10, 11, 14, 16, 23, 27, 28, 29, 31, 32, 38, 47, 53, 59, 64, 67, 71, 79, 83, 86, 100, 103, 107, 114, 125, 127, 128, 131, 136, 137, 139, 147, 149, 151, 167, 170, 172, 173, 176, 179, 191, 197, 199, 202, 211, 223, 227, 229, 235, 239, 251, 256, 263, 265, 269

Offset: 1

Author

Benoit Cloitre, May 10 2002

Keywords

• easy
• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Antti Karttunen)
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
• Wikipedia, Euler's totient function (see the last paragraph in section "Some values of the function").

Crossrefs

Cf. A000005, A000010, A057826, A066412.

Programs

Mathematica

With[{nn = 300}, Function[s, DeleteCases[MapIndexed[If[DivisorSigma[0, First@ #2] == #1, First@ #2, 0] &, Take[#, nn]], 0] &@ Values@ KeySort@ Flatten@ Map[Function[{k, m}, Map[# -> m &, k]] @@ {#, Length@ #} &@ Lookup[s, #] &, Keys@ s]]@ KeySort@ PositionIndex@ Array[EulerPhi, Floor[nn^(4/3)] + 10]] (* Michael De Vlieger, Jul 18 2017 *)

PARI

for(n=1,350,if(sum(i=1,10*n,if(eulerphi(n)-eulerphi(i),0,1))==numdiv(n),print1(n,","))) \\ By the original author. Note: the upper limit 10*n for the search range is quite ad hoc, and is guaranteed to miss some cases when n is large enough. Cf. Wikipedia-article. - Antti Karttunen, Jul 19 2017

PARI

\\ Here is an implementation not using arbitrary limits:
A014197(n, m=1) = { n==1 && return(1+(m<2)); my(p, q); sumdiv(n, d, if( d>=m && isprime(d+1), sum( i=0, valuation(q=n\d, p=d+1), A014197(q\p^i, p))))} \\ M. F. Hasler, Oct 05 2009

PARI

A066412(n) = A014197(eulerphi(n));
isA070305(n) = (A066412(n) == numdiv(n));
n=0; k=1; while(k <= 1000, n=n+1; if(isA070305(n),write("b070305.txt", k, " ", n);k=k+1)); \\ Antti Karttunen, Jul 19 2017

PARI

is(m) = {my(f = factor(m)); invphiNum(eulerphi(f)) == numdiv(f);} \\ Amiram Eldar, Nov 19 2024, using Max Alekseyev's invphi.gp

Scheme

;; With my IntSeq-library.
(define A070305 (MATCHING-POS 1 1 (lambda (n) (= (A066412 n) (A000005 n))))) ;; Antti Karttunen, Jul 18 2017

Formula

Numbers k such that A066412(k) = A000005(k).