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A346694 Primitive terms of A051487.

%I A346694 #39 Mar 31 2023 09:18:20
%S A346694 6,150,726,750,2310,3174,3750,5046,5874,6090,6930,7986,10086,10374,
%T A346694 11550,16854,18270,18750,20790,24378,31122,34650,41334,42630,47526,
%U A346694 54810,57750,62370,63618,64614,73002,76614,87846,93366,93750,102966,103950,127890,140910,146334,146370,164430
%N A346694 Primitive terms of A051487.
%C A346694 If k is an even term greater than 2 of A051487 then 2k is another term.
%C A346694 This sequence lists the initial term k_0 of each infinite subsequence that is solution of the equation phi(k) = phi(k - phi(k)).
%C A346694 About 2: one could argue that 2 is primitive since it is not the double of any previous term of A051487, but as 2^k is not solution for n>1, 2 is not primitive.
%C A346694 Each k_0 is of the form k_0 = 6*m with m odd.
%C A346694 If p > 3 is a Sophie Germain prime, then every m = 2*3*p^q, q >=2 is a term because phi(m) = phi(m-phi(m)) = 2*(p-1)*p^(q-1); the first terms that are not of this form are 6, 2310, 5874, ... (see examples).
%D A346694 Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B42, p. 150.
%e A346694 a(1) = 6 because every k = 3*2^m, m >= 1 satisfies phi(k) = phi(k-phi(k)) = 2^m, and k_0 = 6 is the smallest term of this subsequence of A051487.
%e A346694 a(2) = 150 because every k = 3*5^2*2^m, m >= 1 satisfies phi(k) = phi(k-phi(k)) = 5*2^(m+2) and k_0 = 150 is the smallest term of this subsequence of A051487.
%e A346694 a(3) = 726 because every k = 3*11^2*2^m, m >= 1 satisfies phi(k) = phi(k-phi(k)) = 5*11*2^(m+1) and k_0 = 726 is the smallest term of this subsequence of A051487.
%e A346694 a(5) = 2310 because every k = 3*5*7*11*2^m, m >= 1 satisfies phi(k) = phi(k-phi(k)) = 3*5*2^(m+4) and k_0 = 2310 is the smallest term of this subsequence of A051487.
%p A346694 with(numtheory):
%p A346694 for q from 0 to 13800 do
%p A346694 m := 6*(2*q+1);
%p A346694 if phi(m) = phi(m-phi(m)) then print(m); else fi; od:
%o A346694 (PARI) isdouble(n, list)= {my(v = Vecrev(list)); for(k=1, #v, if (n == 2*v[k], return(1)););}
%o A346694 lista(nn) = {my(list = List(), listp = List()); for (n=3, nn, if (eulerphi(n) == eulerphi(n - eulerphi(n)), if (!isdouble(n, list), listput(listp, n)); listput(list, n););); Vec(listp);} \\ _Michel Marcus_, Aug 06 2021
%Y A346694 Subsequence of A051487.
%Y A346694 Cf. A000010, A051488, A051953, A346692.
%K A346694 nonn
%O A346694 1,1
%A A346694 _Bernard Schott_, Aug 06 2021