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A175177
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Conjectured number of numbers for which the iteration x -> phi(x) + 1 terminates at prime(n). Cardinality of rooted tree T_p (where p is n-th prime) in Karpenko's book.
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5
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2, 3, 4, 9, 2, 31, 6, 4, 2, 2, 2, 11, 24, 41, 2, 2, 2, 57, 2, 2, 58, 2, 2, 6, 17, 4, 2, 2, 39, 67, 2, 2, 2, 2, 2, 2, 25, 4, 2, 2, 2, 158, 2, 61, 2, 2, 2, 2, 2, 2, 54, 2, 186, 2, 10, 2, 2, 2, 18, 8, 2, 2, 2, 2, 96, 2, 2, 18, 2, 6, 15, 2, 2, 2, 2, 2, 2, 44, 34, 6, 2, 16, 2, 105, 2, 2, 60, 5, 4, 2, 2, 2, 4
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OFFSET
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1,1
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, Third Edition, Springer, New York 2004. Chapter B41, Iterations of phi and sigma, page 148.
A. S. Karpenko, Lukasiewicz's Logics and Prime Numbers, (English translation), 2006. See Table 2 on p.125 ff.
A. S. Karpenko, Lukasiewicz's Logics and Prime Numbers, (Russian), 2000.
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LINKS
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EXAMPLE
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a(3) = 4 because x = { 5, 8, 10, 12 } are the 4 numbers from which the iteration x -> phi(x) + 1 terminates at prime(3) = 5.
a(4) = 8 because x = { 7, 9, 14, 15, 16, 18, 20, 24, 30 } are the 9 numbers from which the iteration x -> phi(x) + 1 terminates at prime(4) = 7.
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PROG
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(PARI)
iterat(x) = {my(k, s); if ( isprime(x), return(x)); s=x;
for (k=1, 1000000000, s=eulerphi(s)+1; if(isprime(s), return(s)));
return(s); }
check(y, endrange) = {my(count, start); count=0;
for(start=1, endrange, if(iterat(start)==y, count++; ));
return(count); }
for (n=1, 93, x=prime(n); print1(check(x, 1000000), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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