A346694 Primitive terms of A051487.

6, 150, 726, 750, 2310, 3174, 3750, 5046, 5874, 6090, 6930, 7986, 10086, 10374, 11550, 16854, 18270, 18750, 20790, 24378, 31122, 34650, 41334, 42630, 47526, 54810, 57750, 62370, 63618, 64614, 73002, 76614, 87846, 93366, 93750, 102966, 103950, 127890, 140910, 146334, 146370, 164430

Offset: 1

Author

Bernard Schott, Aug 06 2021

Keywords

• nonn

Comments

If k is an even term greater than 2 of A051487 then 2k is another term.

This sequence lists the initial term k_0 of each infinite subsequence that is solution of the equation phi(k) = phi(k - phi(k)).

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.

Each k_0 is of the form k_0 = 6*m with m odd.

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).

Examples

			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.
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.
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.
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.
		

References

• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B42, p. 150.

Crossrefs

Subsequence of A051487.

Cf. A000010, A051488, A051953, A346692.

Programs

Maple

with(numtheory):
for q from 0 to 13800 do
m := 6*(2*q+1);
if phi(m) = phi(m-phi(m)) then print(m); else fi; od:

PARI

isdouble(n, list)= {my(v = Vecrev(list)); for(k=1, #v, if (n == 2*v[k], return(1)););}
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