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A346694
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Primitive terms of A051487.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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).
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B42, p. 150.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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