cp's OEIS Frontend

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

%I A070305 #54 Mar 04 2025 11:09:14
%S A070305 2,4,8,10,11,14,16,23,27,28,29,31,32,38,47,53,59,64,67,71,79,83,86,
%T A070305 100,103,107,114,125,127,128,131,136,137,139,147,149,151,167,170,172,
%U A070305 173,176,179,191,197,199,202,211,223,227,229,235,239,251,256,263,265,269
%N A070305 Numbers m such that Card(k>0 : phi(k)=phi(m)) = tau(m).
%H A070305 Amiram Eldar, <a href="/A070305/b070305.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Antti Karttunen)
%H A070305 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%H A070305 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 A070305 Numbers k such that A066412(k) = A000005(k).
%t A070305 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 A070305 (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 A070305 (PARI)
%o A070305 \\ Here is an implementation not using arbitrary limits:
%o A070305 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 A070305 (PARI)
%o A070305 A066412(n) = A014197(eulerphi(n));
%o A070305 isA070305(n) = (A066412(n) == numdiv(n));
%o A070305 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 A070305 (PARI) is(m) = {my(f = factor(m)); invphiNum(eulerphi(f)) == numdiv(f);} \\ _Amiram Eldar_, Nov 19 2024, using _Max Alekseyev_'s invphi.gp
%o A070305 (Scheme)
%o A070305 ;; With my IntSeq-library.
%o A070305 (define A070305 (MATCHING-POS 1 1 (lambda (n) (= (A066412 n) (A000005 n))))) ;; _Antti Karttunen_, Jul 18 2017
%Y A070305 Cf. A000005, A000010, A057826, A066412.
%K A070305 easy,nonn
%O A070305 1,1
%A A070305 _Benoit Cloitre_, May 10 2002