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A306771 Numbers m such that m = i + j = i * k and phi(m) = phi(i) + phi(j) = phi(i) * phi(k) for some i, j, k, where phi is the Euler totient function A000010.

%I A306771 #62 Mar 08 2025 06:01:00
%S A306771 3,15,21,33,39,51,57,69,75,87,93,105,111,123,129,141,147,159,165,177,
%T A306771 183,195,201,213,219,231,237,249,255,267,273,285,291,303,309,321,327,
%U A306771 339,345,357,363,375,381,393,399,411,417,429,435,447,453,465,471,483,489
%N A306771 Numbers m such that m = i + j = i * k and phi(m) = phi(i) + phi(j) = phi(i) * phi(k) for some i, j, k, where phi is the Euler totient function A000010.
%C A306771 The 55 terms given in the data section are consistent with a definition "numbers congruent to 3 or 15 mod 18". - _Peter Munn_, May 12 2020
%C A306771 The observation above is true for the first 10^4 terms. - _Amiram Eldar_, Dec 08 2020
%C A306771 The observation above is true for every term; see link. - _Flávio V. Fernandes_, Apr 18 2022
%C A306771 A001748 \ {6, 9} is a subsequence because, for p prime >= 5, 3 * p = p + 2p = p * 3 and phi(3p) = phi(p) + phi(2p) = phi(p) * phi(3) = 2 * (p-1). - _Bernard Schott_, May 13 2022
%H A306771 Amiram Eldar, <a href="/A306771/b306771.txt">Table of n, a(n) for n = 1..10000</a>
%H A306771 Flávio V. Fernandes, <a href="/A306771/a306771.txt">A306771(n) equals 3 times A007310(n)</a>
%H A306771 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F A306771 From _Chai Wah Wu_, Mar 07 2025: (Start)
%F A306771 a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3.
%F A306771 G.f.: x*(3*x^2 + 12*x + 3)/((x - 1)^2*(x + 1)). (End)
%F A306771 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)) (A381671). - _Amiram Eldar_, Mar 08 2025
%e A306771 33 is in the sequence because:
%e A306771 phi(33) = phi(11 + 22) = phi(11) + phi(22) = 10 + 10 = 20, and
%e A306771 phi(33) = phi(3 * 11) = phi(3) * phi(11) = 2 * 10 = 20.
%p A306771 with(numtheory):
%p A306771 for n from 1 to 500 do:
%p A306771 ii:=0:
%p A306771   for i from 1 to trunc(n/2) while(ii=0) do:
%p A306771    if phi(i)+ phi(n-i)= phi(n) and n/i = floor(n/i)
%p A306771       and phi(i)*phi(n/i)=phi(n)
%p A306771       then
%p A306771       ii:=1:printf(`%d, `,n):
%p A306771       else
%p A306771    fi:
%p A306771   od:
%p A306771 od:
%t A306771 LinearRecurrence[{1, 1, -1}, {3, 15, 21}, 100] (* _Paolo Xausa_, Mar 07 2025 *)
%o A306771 (PARI) isok(m) = {my(phim = eulerphi(m)); for (i=1, m\2, if ((eulerphi(i) + eulerphi(m-i) == phim) && !frac(m/i) && (eulerphi(m/i)*eulerphi(i) == phim), return (1)););} \\ _Michel Marcus_, Mar 09 2019
%Y A306771 Cf. A000010, A000040, A007310, A001748, A381671.
%K A306771 nonn,easy
%O A306771 1,1
%A A306771 _Michel Lagneau_, Mar 09 2019
%E A306771 Incorrect comment deleted by _Peter Munn_, May 12 2020
%E A306771 Name corrected by _Flávio V. Fernandes_, Aug 26 2021 and _Peter Munn_, Sep 03 2021