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Numbers m such that Card(k>0 : phi(k)=phi(m)) = tau(m).
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%I #50 Nov 20 2024 07:40:15

%S 2,4,8,10,11,14,16,23,27,28,29,31,32,38,47,53,59,64,67,71,79,83,86,

%T 100,103,107,114,125,127,128,131,136,137,139,147,149,151,167,170,172,

%U 173,176,179,191,197,199,202,211,223,227,229,235,239,251,256,263,265,269

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

%H Amiram Eldar, <a href="/A070305/b070305.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Antti Karttunen)

%H Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Euler&#39;s_totient_function">Euler's totient function</a> (see the last paragraph in section "Some values of the function").

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

%t 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 *)

%o (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

%o (PARI)

%o \\ Here is an implementation not using arbitrary limits:

%o 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

%o (PARI)

%o A066412(n) = A014197(eulerphi(n));

%o isA070305(n) = (A066412(n) == numdiv(n));

%o 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

%o (PARI) is(m) = {my(f = factor(m)); invphiNum(eulerphi(f)) == numdiv(f);} \\ _Amiram Eldar_, Nov 19 2024, using _Max Alekseyev_'s invphi.gp

%o (Scheme, with my IntSeq-library) (define A070305 (MATCHING-POS 1 1 (lambda (n) (= (A066412 n) (A000005 n))))) ;; _Antti Karttunen_, Jul 18 2017

%Y Cf. A000005, A000010, A057826, A066412.

%K easy,nonn,changed

%O 1,1

%A _Benoit Cloitre_, May 10 2002