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A294618
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a(n) is the number of solutions of x^2 = eulerphi(x * m) where x is A293928(n).
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0
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2, 2, 3, 1, 4, 2, 5, 1, 1, 4, 6, 3, 3, 5, 1, 7, 6, 4, 1, 7, 1, 3, 1, 8, 10, 5, 1, 1, 9, 3, 8, 4, 1, 9, 1, 13, 1, 7, 4, 3, 1, 12, 5, 14, 1, 7, 1, 1, 2, 10, 2, 18, 1, 1, 1, 9, 9, 3, 1, 5, 1, 14, 7, 22, 3, 1
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OFFSET
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1,1
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COMMENTS
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The valid values of m in the equation are the terms of the sequence A151999 in order.
m is a solution if all squarefree divisors of x also divide m.
The formula is recursive. For example, taking a151999(68) we get the following: 11664=phi(108*324), 1259712=phi(11664*324), 136048896=phi(1259712*324), ...
If a solution exists then x^(k+1) = phi(x^k * m) for a fixed m, and the smallest value of k must be 1. This follows from a|b implies phi(a)|phi(b), and for k >= 1 a^(k-1)|a^k.
The smallest solution where solutions exist are the terms of the sequence A055744 not in order.
The values of phi(m) are the terms of the sequence A068997 not in order.
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LINKS
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FORMULA
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0 < (phi(m)^(k+1) = phi(phi(m)^k*m)), k >= 1, m >= 1.
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EXAMPLE
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The first 1 is a term since there is only 1 solution when phi(m)=6. The solution is m=18.
The first 5 is a term since there are 5 solutions when phi(m)=16. These are 32, 34, 40, 48, and 60.
Illustration of first few terms:
1: [1, 2],
2: [4, 6],
4: [8, 10, 12],
6: [18],
8: [16, 20, 24, 30],
12: [36, 42],
16: [32, 34, 40, 48, 60],
18: [54],
20: [50],
24: [72, 78, 84, 90],
32: [64, 68, 80, 96, 102, 120],
... (End)
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PROG
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(PARI) isok(n) = {iv = invphi(n); if (#iv, return (sum(m=1, #iv, n^2 == eulerphi(n*iv[m])))); return (0); }
lista(nn) = {for (n=1, nn, if (v = isok(n), print1(v, ", ")); ); } \\ \\ using the invphi script by Max Alekseyev; Michel Marcus, Nov 07 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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